Combinatorics

Problem 1

Tom has 4 different marbles and 3 identical yellow ones. How many unique pairs of marbles can he select?

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Problem 2

Tom has 4 unique marbles and 3 identical yellow ones. How many ways can he choose 2 marbles?

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Problem 3

How many unique 4-letter passwords can be formed from the 26-letter English alphabet without repeating letters? [?] ways

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Problem 4

How many ways can you choose 3 books from 11? Use the formula for combinations: $\binom{11}{3}$ .

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Problem 5

How many ways can you play 3 songs from 8 using random shuffle? Use ${ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}$ .

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Problem 6

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use ${ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}$ .

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Problem 7

How many ways can you choose 5 numbers from 1 to 59 and 1 bonus number from 1 to 35?

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Problem 8

How many ways can 10 athletes finish in the top 3 positions of a $400 \mathrm{~m}$ race? Options: a. 720 b. 120 c. $10^{3}$ d. 3

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Problem 9

A college conference has 3 presenters in Hall A, 2 in Hall B, and 4 in Hall C. How many ways to attend 2 from C and 1 from A or B?

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Problem 10

How many ways can you attend 2 presentations in Hall C and 1 in either Hall A or B if there are 3 in A, 2 in B, and 4 in C?

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Problem 11

a.) ¿Cuántas delegaciones diferentes se pueden formar con al menos un miembro de 18 estudiantes? b.) ¿Cuántas delegaciones diferentes se pueden formar con al menos dos miembros de 18 estudiantes?

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Problem 12

หาจำนวนรหัส 4 ตัวที่สร้างจากพยัญชนะ ก, ข, ค, ง โดยไม่ซ้ำกัน (24)

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Problem 13

11. หาเลข 3 หลักคู่ที่ใช้เลขซ้ำและไม่ซ้ำได้ทั้งหมด $(450,328)$
12. หาวิธีนั่ง 3 คู่สามีภรรยา โดยผู้ชายต้องนั่งปลายแถวเสมอ (144)
13. จัดหลอดไฟสีแดง 3 หลอด, เหลือง 4 หลอด, น้ำเงิน 2 หลอด ใน 9 ขั้ว (1,260)
14. จัดแถว 5 คนจากห้อง ก, 3 คนจากห้อง ข, 2 คนจากห้อง ค โดยนักเรียนห้องเดียวกันติดกัน (8,640)

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Problem 14

หาเลข 3 หลักคู่ที่มีเลขซ้ำและไม่ซ้ำกัน $(450,328)$ และวิธีนั่งของสามีภรรยาที่ผู้ชายต้องนั่งปลายแถว (144)

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Problem 15

Arrange 8 students with the following conditions: 1) A & B together: 10,080; 2) A & B at ends: 1,440; 3) A & B apart: 30,240; 4) Alternating genders: 1,152; 5) 4 boys & 4 girls together: 1,152.

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Problem 16

Identify the correct notation for permutation from these options: $\mathrm{rPn}$ , $P(n, r)$ , $C(n, r)$ , $\mathrm{nCr}$ .

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Problem 17

What is the probability of guessing a 5-digit code from 5 unique numbers? A. $\frac{1}{12}$ B. $\frac{1}{120}$ C. $\frac{1}{40}$ D. $\frac{1}{60}$

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Problem 18

How many ways can you use nickels, dimes, and quarters to pay for a 65-cent item?

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Problem 19

Un concurso eligió 94 funcionarios que hablan inglés, francés y alemán. ¿Cuántos hablan al menos uno y al menos dos idiomas?

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Problem 20

How many ways can you use nickels, dimes, and quarters to pay for a 65-cent item? Answer: 18 ways.

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Problem 21

En un total de 90 aspirantes, 40 quieren Derecho, 38 Enfermería, 33 Gastronomía. ¿Cuántos desean al menos uno y al menos dos programas?

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Problem 22

En una universidad, 90 aspirantes quieren estudiar Derecho (40), Enfermería (38) y Gastronomía (33). Hay intersecciones: 20 Derecho y Enfermería, 18 Derecho y Gastronomía, 14 Enfermería y Gastronomía, y 9 en los tres.
a. ¿Cuántos aspirantes quieren al menos uno de los programas? b. ¿Cuántos quieren al menos dos programas?

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Problem 23

How many handshakes occur when 4 people each shake hands once? Generalize for $n$ people. Simplify your answer.

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Problem 24

How many handshakes occur when 4 people shake hands once? Generalize for $n$ people.

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Problem 25

How many handshakes occur when 4 people shake hands once? Generalize for $n$ people.

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Problem 26

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9:
8 & & 9 & 7 & 1 & & 6
Choose from options A, B, C, or D.

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Problem 27

In a league of 6 teams where each team plays every other team 3 times, how many total games are there? Use $G = 3 \times \binom{6}{2}$ .

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Problem 28

a.) ¿Cuántas delegaciones diferentes se pueden formar con al menos 1 miembro de 16 estudiantes? b.) ¿Cuántas delegaciones diferentes se pueden formar con al menos 2 miembros de 16 estudiantes?

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Problem 29

Find the number of subsets and proper subsets of the set $A=\{8,9,10,11,12,14,15,16\}$ .

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Problem 30

How many ways can a team of 15 select a captain and a vice captain? Options: a. 30 b. 105 c. 210 d. 225

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Problem 31

How many ways can 1st, 2nd, and 3rd places be awarded among 6 competing bands? Options: a. 120 b. 30 c. 6 d. 20

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Problem 32

How many ways can you choose a president and vice-president from 9 people? Use the formula for permutations: $P(n, r) = \frac{n!}{(n-r)!}$ .

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Problem 33

Calculate the number of ways to choose a president and vice-president from 7 people.

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Problem 34

How many ways can you elect a president and vice-president from 8 people? Use the formula for permutations: $P(n, r) = \frac{n!}{(n-r)!}$ .

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Problem 35

How many ways can you choose a president and vice-president from 7 people? Calculate using permutations: $P(7, 2)$ .

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Problem 36

When will birds (every 9 minutes) and dancers (every 15 minutes) pop out together again in the next 60 minutes?

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Problem 37

How many different salad variations can be made with lettuce and any combination of green peppers, tomatoes, sunflower seeds, cucumbers, banana peppers, carrots, and mushrooms?

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Problem 38

Find the number of ways to choose 2 distinct items from 10 food items. Options: a. 5 b. 20 c. 25 d. 45

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Problem 39

Determine which scenario requires permutations to count the arrangements: A, B, C, or D.

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Problem 40

A group has 5 men and 7 women. 6 people are chosen.
a. How many ways to select 6 from 12? b. How many ways to select 6 women from 7? c. What is the probability all selected are women? a. Ways to select 6 from 12 is $\binom{12}{6}$ .

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Problem 41

A group has 7 men and 6 women. Select 3 people. Find: a. Ways to choose 3 from 13. b. Ways to choose 3 women from 6. c. Probability all selected are women. a. The number of ways to select is $\binom{13}{3}$ .

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Problem 42

Find the probabilities for the following scenarios with 6 comics (A, B, C, D, E, F): a. B first, b. F second & A fourth, c. order B, F, A, C, D, E, d. D or E fifth.

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Problem 43

Find the probability of selecting 2 Democrats and 2 Republicans from a committee of 4 chosen from 8 Democrats and 5 Republicans.

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Problem 44

Calculate the number of diamond flushes in a 3-card poker hand and its probability. Also, find total 3-card hands.

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Problem 45

A hand has 5 cards from a 52-card deck. Find: a. Total 5-card hands, b. Heart flush hands, c. Probability of a heart flush.

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Problem 46

In a talent show with 8 acts, find the probabilities of: A) singer first, comedian second, dancer third, pianist fourth; B) any order of pianist, comedian, dancer, juggler in first four. Provide answers as simplified fractions. $P(A)=$ $P(B)=$

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Problem 47

You and your brother work from 4 PM to midnight. You have every 9th night off, he has every 6th. When is your next shared night off after April 1?

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Problem 48

How many pairwise comparisons are needed to find a winner among 3 candidates? (Type an integer.)

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Problem 49

How many pairwise comparisons are needed to find a winner among 10 candidates in an election? (Type an integer.)

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Problem 50

Find the number of ways to wear one bracelet and one necklace from 6 bracelets and 15 necklaces.

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Problem 51

Find the number of ways to arrange 6 train cars after an engine. (Engine is first.) Answer: $6!$ ways.

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Problem 52

What is the probability that the sum of four randomly chosen numbers from $\{1,2,3,\ldots,10\}$ is odd?

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Problem 53

Find the total subsets of the set $\{12, 13, 14\}$ .

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Problem 54

Find the total number of subsets of the set $\{1,2,3, \ldots, 8\}$ .

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Problem 55

In a checker puzzle, move 1 checker from blue to red side, switching colors. What’s the minimum moves needed?

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Problem 56

In a checker game, move colors to switch sides. 1. With 1 checker each, how many moves for blue right, red left? $3 \text{ moves}$ 2. With 3 checkers each, find moves needed. 3. Estimate for 2 or 4 checkers, then test. 4. Explain how 3 checkers help with 4. 5. Estimate moves for 7 checkers each.

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Problem 57

A woman has 3 hats, 2 jackets, and 6 blouses. How many unique outfits can she create with one of each?

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Problem 58

How many different car choices can you make with 3 body styles, 3 colors, and 6 models?

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Problem 59

How many different car choices are there with 3 body styles, 3 colors, and 6 models?

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Problem 60

A pianist has 3 pieces to arrange for a recital. How many different arrangements are possible?

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Problem 61

A pianist can arrange 3 pieces in how many different ways? Answer: $3!$

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Problem 62

In an 800 meter race with 7 runners, how many ways can they finish 1st, 2nd, and 3rd?

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Problem 63

How many ways can 4 positions (President, VP, Secretary, Treasurer) be filled from 11 students?

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Problem 64

In how many ways can 7 runners place 1st, 2nd, and 3rd in an 800 meter race?

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Problem 65

Evaluate $C(7,3)$ or ${ }_{7} C_{3}$ . What is the result?

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Problem 66

How many ways can you choose 3 students from a class of 18 for a field trip?

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Problem 67

Find the probabilities for a 5-card hand from a 52-card deck:
a. Exactly 2 kings: $P(2 \text{ Kings})=0.0399$ b. All hearts: $P(\text{ All hearts})=0.000495$ c. Exactly 4 face cards: $P(4 \text{ Face Cards})=$

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Problem 68

How many distinct arrangements can be made with the letters in "ROBBERS"?

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Problem 69

How many unique arrangements can be made from the letters in "PROGRAMMING"? There are $\square$ distinct permutations.

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Problem 70

How many ways can six city commissioners be selected from fourteen candidates? Use the combination formula: $\binom{14}{6}$ .

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Problem 71

Find the number of distinct arrangements of the digits in $8,663,333$ .

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Problem 72

Find the number of committees of 2 teachers and 3 students from 6 teachers and 41 students. Answer: 3003.

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Problem 73

How many ways can you choose 5 books from 18 available books for your vacation? Use the formula for combinations: $\binom{18}{5}$ .

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Problem 74

How many ways can you form a committee of 3 teachers and 4 students from 9 teachers and 43 students?

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Problem 75

How many ways can you select all 48 items from a sample space without replacement? Express your answer in factorial notation.

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Problem 76

How many ways can 3 out of 12 bands be chosen for the top time slots? Use combinations: $C(12,3)$ .

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Problem 77

How many ways can you draw 100 unique names from a hat without replacement? Write your answer in factorial notation: $100!$

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Problem 78

How many unique patterns of 4 colors can be formed using the colors red, orange, yellow, green, blue, and violet?

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Problem 79

How many ways can you choose 2 different instructors from 6 for your statistics classes?

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Problem 80

How many ways can players choose 5 unique symbols from 9 total symbols for a password? Answer in simplest form: $\binom{9}{5}$ .

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Problem 81

How many ways can you choose 2 different instructors from 6 for your statistics classes? Use the formula for combinations: $\binom{6}{2}$ .

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Problem 82

How many unique 5-symbol passwords can be formed from 9 different symbols? Provide your answer in simplest form.

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Problem 83

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions?

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Problem 84

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions? Use the formula: $P(13, 5)$ .

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Problem 85

How many ways can nursing students choose 5 classes from 11 without repeating any class?

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Problem 86

How many ways can 6 people sit in 10 seats? Provide your answer in simplest form.

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Problem 87

How many ways can 35 new employees be grouped into sets of 5 without repetition? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

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Problem 88

How many ways can 11 out of 15 representatives be chosen for presentations? Calculate using combinations: $\binom{15}{11}$ .

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Problem 89

How many ways can 11 out of 15 representatives be chosen for presentations? Use the formula for combinations: $\binom{15}{11}$ .

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Problem 90

How many ways can a customer choose 13 different types of doughnuts from 21 available types? Use combinations: $\binom{21}{13}$ .

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Problem 91

Calculate the total combinations of a password with 3 digits ( $10$ options) and 8 letters ( $26$ options) without repetition.

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Problem 92

A bookseller selects 20 of 30 fiction and 10 of 25 nonfiction books. Calculate combinations using $C(n, k) = \frac{n!}{k!(n-k)!}$ .

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Problem 93

How many ways can you choose 2 characters from 8 volunteers for a play? Answer: $C(8, 2)$ .

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Problem 94

How many ways can 5 players be selected from 25 for a basketball game? Use the formula for combinations: $\binom{25}{5}$ .

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Problem 95

How many different car choices are there with 3 body styles, 2 colors, and 3 models? Calculate: $3 \times 2 \times 3$ .

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Problem 96

Calculate the number of possible passwords starting with "A" or "B", followed by 6 characters (digits or lower-case letters).

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Problem 97

How many ways can you choose 5 toppings from 11 options at a deli? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

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Problem 98

Calculate the number of ways to award gold, silver, and bronze medals to 8 sprinters in a 100-meter race.

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Problem 99

A pianist has 8 pieces for a recital. How many ways can she arrange them? Your answer is: $8!$

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Problem 100

In how many ways can 8 runners finish in 1st, 2nd, and 3rd places in an 800 meter race?

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Combinatorics

Problem 1

Tom has 4 different marbles and 3 identical yellow ones. How many unique pairs of marbles can he select?

Problem 2

Tom has 4 unique marbles and 3 identical yellow ones. How many ways can he choose 2 marbles?

Problem 3

How many unique 4-letter passwords can be formed from the 26-letter English alphabet without repeating letters? [?] ways

Problem 4

How many ways can you choose 3 books from 11? Use the formula for combinations: (113)\binom{11}{3}(311​).

Problem 5

How many ways can you play 3 songs from 8 using random shuffle? Use 8P3=8!(8−3)!{ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}8​P3​=(8−3)!8!​.

Problem 6

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use 100C3=100!3!(100−3)!{ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}100​C3​=3!(100−3)!100!​.

Problem 7

How many ways can you choose 5 numbers from 1 to 59 and 1 bonus number from 1 to 35?

Problem 8

How many ways can 10 athletes finish in the top 3 positions of a 400 m400 \mathrm{~m}400 m race? Options: a. 720 b. 120 c. 10310^{3}103 d. 3

Problem 9

A college conference has 3 presenters in Hall A, 2 in Hall B, and 4 in Hall C. How many ways to attend 2 from C and 1 from A or B?

Problem 10

How many ways can you attend 2 presentations in Hall C and 1 in either Hall A or B if there are 3 in A, 2 in B, and 4 in C?

Problem 11

a.) ¿Cuántas delegaciones diferentes se pueden formar con al menos un miembro de 18 estudiantes? b.) ¿Cuántas delegaciones diferentes se pueden formar con al menos dos miembros de 18 estudiantes?

Problem 12

หาจำนวนรหัส 4 ตัวที่สร้างจากพยัญชนะ ก, ข, ค, ง โดยไม่ซ้ำกัน (24)

Problem 13

Problem 14

หาเลข 3 หลักคู่ที่มีเลขซ้ำและไม่ซ้ำกัน (450,328)(450,328)(450,328) และวิธีนั่งของสามีภรรยาที่ผู้ชายต้องนั่งปลายแถว (144)

Problem 15

Arrange 8 students with the following conditions: 1) A & B together: 10,080; 2) A & B at ends: 1,440; 3) A & B apart: 30,240; 4) Alternating genders: 1,152; 5) 4 boys & 4 girls together: 1,152.

Problem 16

Identify the correct notation for permutation from these options: rPn\mathrm{rPn}rPn, P(n,r)P(n, r)P(n,r), C(n,r)C(n, r)C(n,r), nCr\mathrm{nCr}nCr.

Problem 17

What is the probability of guessing a 5-digit code from 5 unique numbers? A. 112\frac{1}{12}121​ B. 1120\frac{1}{120}1201​ C. 140\frac{1}{40}401​ D. 160\frac{1}{60}601​

Problem 18

How many ways can you use nickels, dimes, and quarters to pay for a 65-cent item?

Problem 19

Un concurso eligió 94 funcionarios que hablan inglés, francés y alemán. ¿Cuántos hablan al menos uno y al menos dos idiomas?

Problem 20

How many ways can you use nickels, dimes, and quarters to pay for a 65-cent item? Answer: 18 ways.

Problem 21

En un total de 90 aspirantes, 40 quieren Derecho, 38 Enfermería, 33 Gastronomía. ¿Cuántos desean al menos uno y al menos dos programas?

Problem 22

Problem 23

How many handshakes occur when 4 people each shake hands once? Generalize for nnn people. Simplify your answer.

Problem 24

How many handshakes occur when 4 people shake hands once? Generalize for nnn people.

Problem 25

How many handshakes occur when 4 people shake hands once? Generalize for nnn people.

Problem 26

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9: 8 & & 9 & 7 & 1 & & 6 Choose from options A, B, C, or D.

Problem 27

In a league of 6 teams where each team plays every other team 3 times, how many total games are there? Use G=3×(62)G = 3 \times \binom{6}{2}G=3×(26​).

Problem 28

a.) ¿Cuántas delegaciones diferentes se pueden formar con al menos 1 miembro de 16 estudiantes? b.) ¿Cuántas delegaciones diferentes se pueden formar con al menos 2 miembros de 16 estudiantes?

Problem 29

Find the number of subsets and proper subsets of the set A={8,9,10,11,12,14,15,16}A=\{8,9,10,11,12,14,15,16\}A={8,9,10,11,12,14,15,16}.

Problem 30

How many ways can a team of 15 select a captain and a vice captain? Options: a. 30 b. 105 c. 210 d. 225

Problem 31

How many ways can 1st, 2nd, and 3rd places be awarded among 6 competing bands? Options: a. 120 b. 30 c. 6 d. 20

Problem 32

How many ways can you choose a president and vice-president from 9 people? Use the formula for permutations: P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n-r)!}P(n,r)=(n−r)!n!​.

Problem 33

Calculate the number of ways to choose a president and vice-president from 7 people.

Problem 34

How many ways can you elect a president and vice-president from 8 people? Use the formula for permutations: P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n-r)!}P(n,r)=(n−r)!n!​.

Problem 35

How many ways can you choose a president and vice-president from 7 people? Calculate using permutations: P(7,2)P(7, 2)P(7,2).

Problem 36

When will birds (every 9 minutes) and dancers (every 15 minutes) pop out together again in the next 60 minutes?

Problem 37

How many different salad variations can be made with lettuce and any combination of green peppers, tomatoes, sunflower seeds, cucumbers, banana peppers, carrots, and mushrooms?

Problem 38

Find the number of ways to choose 2 distinct items from 10 food items. Options: a. 5 b. 20 c. 25 d. 45

Problem 39

Determine which scenario requires permutations to count the arrangements: A, B, C, or D.

Problem 40

A group has 5 men and 7 women. 6 people are chosen. a. How many ways to select 6 from 12? b. How many ways to select 6 women from 7? c. What is the probability all selected are women? a. Ways to select 6 from 12 is (126)\binom{12}{6}(612​).

Problem 41

How many ways can you choose 3 books from 11? Use the formula for combinations: $\binom{11}{3}$ .

How many ways can you play 3 songs from 8 using random shuffle? Use ${ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}$ .

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use ${ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}$ .

How many ways can 10 athletes finish in the top 3 positions of a $400 \mathrm{~m}$ race? Options: a. 720 b. 120 c. $10^{3}$ d. 3

หาเลข 3 หลักคู่ที่มีเลขซ้ำและไม่ซ้ำกัน $(450,328)$ และวิธีนั่งของสามีภรรยาที่ผู้ชายต้องนั่งปลายแถว (144)

Identify the correct notation for permutation from these options: $\mathrm{rPn}$ , $P(n, r)$ , $C(n, r)$ , $\mathrm{nCr}$ .

What is the probability of guessing a 5-digit code from 5 unique numbers? A. $\frac{1}{12}$ B. $\frac{1}{120}$ C. $\frac{1}{40}$ D. $\frac{1}{60}$

How many handshakes occur when 4 people each shake hands once? Generalize for $n$ people. Simplify your answer.

How many handshakes occur when 4 people shake hands once? Generalize for $n$ people.

How many handshakes occur when 4 people shake hands once? Generalize for $n$ people.

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9:
8 & & 9 & 7 & 1 & & 6
Choose from options A, B, C, or D.

In a league of 6 teams where each team plays every other team 3 times, how many total games are there? Use $G = 3 \times \binom{6}{2}$ .

Find the number of subsets and proper subsets of the set $A=\{8,9,10,11,12,14,15,16\}$ .

How many ways can you choose a president and vice-president from 9 people? Use the formula for permutations: $P(n, r) = \frac{n!}{(n-r)!}$ .

How many ways can you elect a president and vice-president from 8 people? Use the formula for permutations: $P(n, r) = \frac{n!}{(n-r)!}$ .

How many ways can you choose a president and vice-president from 7 people? Calculate using permutations: $P(7, 2)$ .

A group has 5 men and 7 women. 6 people are chosen.
a. How many ways to select 6 from 12? b. How many ways to select 6 women from 7? c. What is the probability all selected are women? a. Ways to select 6 from 12 is $\binom{12}{6}$ .

A group has 7 men and 6 women. Select 3 people. Find: a. Ways to choose 3 from 13. b. Ways to choose 3 women from 6. c. Probability all selected are women. a. The number of ways to select is $\binom{13}{3}$ .

In a talent show with 8 acts, find the probabilities of: A) singer first, comedian second, dancer third, pianist fourth; B) any order of pianist, comedian, dancer, juggler in first four. Provide answers as simplified fractions. $P(A)=$ $P(B)=$

Find the number of ways to arrange 6 train cars after an engine. (Engine is first.) Answer: $6!$ ways.

What is the probability that the sum of four randomly chosen numbers from $\{1,2,3,\ldots,10\}$ is odd?

Find the total subsets of the set $\{12, 13, 14\}$ .

Find the total number of subsets of the set $\{1,2,3, \ldots, 8\}$ .

A pianist can arrange 3 pieces in how many different ways? Answer: $3!$

Evaluate $C(7,3)$ or ${ }_{7} C_{3}$ . What is the result?

How many unique arrangements can be made from the letters in "PROGRAMMING"? There are $\square$ distinct permutations.

How many ways can six city commissioners be selected from fourteen candidates? Use the combination formula: $\binom{14}{6}$ .

Find the number of distinct arrangements of the digits in $8,663,333$ .

How many ways can you choose 5 books from 18 available books for your vacation? Use the formula for combinations: $\binom{18}{5}$ .