Question

There are 5 freshmen, 8 sophomores, and 7 juniors in a chess club. A group of 6 students will be chosen to compete in a competition. How many combinations of students are possible if the group is to consist of exactly 3 freshmen?

Answer 1

The number of combinations of students are possible if the group is to consist of exactly 3 freshmen = 176358000

Explanation:

Given : There are 5 freshmen, 8 sophomores, and 7 juniors in a chess club.

Total students =  5+8+7=20

Number of students other than freshmen = 15

A group of 6 students will be chosen to compete in a competition.

The total number of ways to select 6 students out of 20= $^{20}C_{6}$

Number of ways to select exactly 3 freshmen = $^5C_3\times^{15}C_3$

Then , the number of combinations of students are possible if the group is to consist of exactly 3 freshmen :$^{20}C_{6}\times^5C_3\times^{15}C_3$

$=\dfrac{20!}{6!14!}\times\dfrac{5!}{3!\times2!}\times\dfrac{15!}{3!12!}\\\\=176358000$

Hence, the number of combinations of students are possible if the group is to consist of exactly 3 freshmen = 176358000

# Learn more :

There are 5 violinists, 8 guitarists, and 7 drummers in a music club. A group of 6 students will be chosen to compete in a competition. How many combinations of students are possible if the group is to consist of an equal number of violinists, guitarists, and drummers?

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