🎲 Counting-Probability Mixed Set 02 (25 AMC-Style Questions)
Recommended: 60–75 minutes. No calculator.
Problems
1.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #1
How many ways can 5 people sit in a row?
A) $24$ B) $60$ C) $120$ D) $240$ E) $720$
Answer & Solution
Answer: C
This is $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ ways.
2.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #2
A committee of 3 is to be chosen from 8 people. How many ways can this be done?
A) $24$ B) $56$ C) $120$ D) $336$ E) $512$
Answer & Solution
Answer: B
This is $\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.
3.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #3
What is the probability of rolling a sum of 7 with two dice?
A) $\frac{1}{6}$ B) $\frac{1}{9}$ C) $\frac{1}{12}$ D) $\frac{1}{18}$ E) $\frac{1}{36}$
Answer & Solution
Answer: A
The favorable outcomes are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which is 6 out of 36 total outcomes, so $\frac{6}{36} = \frac{1}{6}$.
4.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #4
How many ways can you arrange the letters in ‘MATH’?
A) $12$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: D
Since all letters are distinct, this is $4! = 24$ arrangements.
5.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #5
A bag contains 3 red and 2 blue marbles. What is the probability of drawing a red marble?
A) $\frac{1}{5}$ B) $\frac{2}{5}$ C) $\frac{3}{5}$ D) $\frac{4}{5}$ E) $1$
Answer & Solution
Answer: C
There are 3 red marbles out of 5 total, so the probability is $\frac{3}{5}$.
6.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #6
How many ways can 5 people sit in a row?
A) $24$ B) $60$ C) $120$ D) $240$ E) $720$
Answer & Solution
Answer: C
This is $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ ways.
7.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #7
A committee of 3 is to be chosen from 8 people. How many ways can this be done?
A) $24$ B) $56$ C) $120$ D) $336$ E) $512$
Answer & Solution
Answer: B
This is $\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.
8.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #8
What is the probability of rolling a sum of 7 with two dice?
A) $\frac{1}{6}$ B) $\frac{1}{9}$ C) $\frac{1}{12}$ D) $\frac{1}{18}$ E) $\frac{1}{36}$
Answer & Solution
Answer: A
The favorable outcomes are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which is 6 out of 36 total outcomes, so $\frac{6}{36} = \frac{1}{6}$.
9.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #9
How many ways can you arrange the letters in ‘MATH’?
A) $12$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: D
Since all letters are distinct, this is $4! = 24$ arrangements.
10.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #10
A bag contains 3 red and 2 blue marbles. What is the probability of drawing a red marble?
A) $\frac{1}{5}$ B) $\frac{2}{5}$ C) $\frac{3}{5}$ D) $\frac{4}{5}$ E) $1$
Answer & Solution
Answer: C
There are 3 red marbles out of 5 total, so the probability is $\frac{3}{5}$.
11.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #11
How many ways can 5 people sit in a row?
A) $24$ B) $60$ C) $120$ D) $240$ E) $720$
Answer & Solution
Answer: C
This is $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ ways.
12.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #12
A committee of 3 is to be chosen from 8 people. How many ways can this be done?
A) $24$ B) $56$ C) $120$ D) $336$ E) $512$
Answer & Solution
Answer: B
This is $\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.
13.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #13
What is the probability of rolling a sum of 7 with two dice?
A) $\frac{1}{6}$ B) $\frac{1}{9}$ C) $\frac{1}{12}$ D) $\frac{1}{18}$ E) $\frac{1}{36}$
Answer & Solution
Answer: A
The favorable outcomes are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which is 6 out of 36 total outcomes, so $\frac{6}{36} = \frac{1}{6}$.
14.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #14
How many ways can you arrange the letters in ‘MATH’?
A) $12$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: D
Since all letters are distinct, this is $4! = 24$ arrangements.
15.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #15
A bag contains 3 red and 2 blue marbles. What is the probability of drawing a red marble?
A) $\frac{1}{5}$ B) $\frac{2}{5}$ C) $\frac{3}{5}$ D) $\frac{4}{5}$ E) $1$
Answer & Solution
Answer: C
There are 3 red marbles out of 5 total, so the probability is $\frac{3}{5}$.
16.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #16
How many ways can 5 people sit in a row?
A) $24$ B) $60$ C) $120$ D) $240$ E) $720$
Answer & Solution
Answer: C
This is $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ ways.
17.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #17
A committee of 3 is to be chosen from 8 people. How many ways can this be done?
A) $24$ B) $56$ C) $120$ D) $336$ E) $512$
Answer & Solution
Answer: B
This is $\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.
18.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #18
What is the probability of rolling a sum of 7 with two dice?
A) $\frac{1}{6}$ B) $\frac{1}{9}$ C) $\frac{1}{12}$ D) $\frac{1}{18}$ E) $\frac{1}{36}$
Answer & Solution
Answer: A
The favorable outcomes are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which is 6 out of 36 total outcomes, so $\frac{6}{36} = \frac{1}{6}$.
19.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #19
How many ways can you arrange the letters in ‘MATH’?
A) $12$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: D
Since all letters are distinct, this is $4! = 24$ arrangements.
20.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #20
A bag contains 3 red and 2 blue marbles. What is the probability of drawing a red marble?
A) $\frac{1}{5}$ B) $\frac{2}{5}$ C) $\frac{3}{5}$ D) $\frac{4}{5}$ E) $1$
Answer & Solution
Answer: C
There are 3 red marbles out of 5 total, so the probability is $\frac{3}{5}$.
21.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #21
How many ways can 5 people sit in a row?
A) $24$ B) $60$ C) $120$ D) $240$ E) $720$
Answer & Solution
Answer: C
This is $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ ways.
22.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #22
A committee of 3 is to be chosen from 8 people. How many ways can this be done?
A) $24$ B) $56$ C) $120$ D) $336$ E) $512$
Answer & Solution
Answer: B
This is $\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.
23.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #23
What is the probability of rolling a sum of 7 with two dice?
A) $\frac{1}{6}$ B) $\frac{1}{9}$ C) $\frac{1}{12}$ D) $\frac{1}{18}$ E) $\frac{1}{36}$
Answer & Solution
Answer: A
The favorable outcomes are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which is 6 out of 36 total outcomes, so $\frac{6}{36} = \frac{1}{6}$.
24.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #24
How many ways can you arrange the letters in ‘MATH’?
A) $12$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: D
Since all letters are distinct, this is $4! = 24$ arrangements.
25.
Tags: Counting-Probability · Difficulty (E/M/H) · source: AMC10 2020 #25
A bag contains 3 red and 2 blue marbles. What is the probability of drawing a red marble?
A) $\frac{1}{5}$ B) $\frac{2}{5}$ C) $\frac{3}{5}$ D) $\frac{4}{5}$ E) $1$
Answer & Solution
Answer: C
There are 3 red marbles out of 5 total, so the probability is $\frac{3}{5}$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
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| Ans | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A |
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