# Probability That 1, 2, 3, 4 of the Recruits Will Receive the Correct Size of Boots

**Situation**

Four army recruits went to the supply room to get their military boots. Their shoe sizes were 7, 8, 9 & 10. The supply officer, after being informed of their sizes, prepared the four pairs of boots they need. If the boots are handed to each of the four recruits at random, what is the probability that...

- exactly 3 of them will receive the correct shoe size?
A. 1/16 C. 1/12 B. 1/24 D. 0 - all of them will receive the correct shoe size?
A. 1/16 C. 1/12 B. 1/24 D. 0 - none of them will receive the correct shoe size
A. 3/8 C. 1/16 B. 23/24 D. 5/12

**Answer Key**

Part 1: [ D ]

Part 2: [ B ]

Part 3: [ A ]

Part 2: [ B ]

Part 3: [ A ]

**Solution**

Total number of ways,

Exactly 1 is correct,

Exactly 2 are correct,

Exactly 3 are correct,

Exactly 4 are correct,

*N*= 4! = 24Exactly 1 is correct,

*N*_{1}= 2(4C1) = 8Exactly 2 are correct,

*N*_{2}= 4C2 = 6Exactly 3 are correct,

*N*_{3}= 0Exactly 4 are correct,

*N*_{4}= 1**Part 1:** Probability that exactly 3 are correct

$P = \dfrac{N_3}{N} = \dfrac{0}{24}$

$P = 0$ ← *answer*

**Part 2:** Probability that all are correct

$P = \dfrac{N_4}{N}$

$P = 1/24$ ← *answer*

**Part 3:** Probability that none is correct

$P = \dfrac{N - N_1 - N_2 - N_3 - N_4}{N} = \dfrac{24 - 8 - 6 - 0 - 1}{24}$

$P = 3/8$ ← *answer*