[Ezra Halleck]

In a statistics club, there are 7 sophomores, and 3 freshmen.
a) In how many ways can the club form a committee of 3 students without regard to rank?
b) In how many ways can the club form a committee which consists of 2 sophomores and 1 freshman?
c) The club has decided to elect a president, vice president, treasurer and secretary. In how many ways can they do this if the pres and vp must be sophomores and the treasurer and secretary must be freshman?

[Eric Osorio]

a) 10 P 3 = 720 different ways to form a club committee

[Ezra Halleck]

a) is a combination rather than a permutation problem as a “committee” traditionally does not have an ordering although often times there special assignments such as chair or secretary. The pool is 10 and the size is 3, so 10 nCr 3 or 10*9*8/(3*2)=120
b) is a product of combinations, as we are selecting separately from 2 pools. 7 nCr 2 * 3 nCr 1= 7*6/2 * 3=63
c) Here we are working with a product of permutations, one for each pool. Selecting the p and vp: 7 nPr2 and the t and s: 3 nPr 2 or 7*6*3*2=252

[Vitaliy]

B) 7 nCr 2 = 42÷2 = 21
3 nCr 1 = 3
7 nCr 2 × 3 nCr 1 = 21×3 = 63

Great explanation professor. Thanks.