Question

14 students have volunteered for a committee. Eight of them are seniors and six of them are juniors. (a) How many ways are there to select a committee of 5 students

Answer 1

Answer:  2002

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Explanation:

We have five slots to fill, which I'll label A through E.

There are,

• 14 choices for slot A
• 13 choices for slot B
• 12 choices for slot C
• 11 choices for slot D
• 10 choices for slot E

Each time we fill up a slot, we count down by 1.

This gives 14*13*12*11*10 = 240,240 different permutations. If order mattered, then this would be the answer.

However, order doesn't matter on a committee without rankings. Each member has the same rank as any other.

Within any group of 5 people, there are 5! = 5*4*3*2*1 = 120 ways to arrange them. This means we divide that 240,240 figure by 120 to find the number of combinations.

There are (240,240)/(120) = 2002 different combinations.

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Here's how we could have done the problem using the nCr combination formula.

Plug in n = 14 and r = 5.

n C r = (n!)/(r!(n-r)!)

14 C 5 = (14!)/(5!*(14-5)!)

14 C 5 = (14!)/(5!*9!)

14 C 5 = (14*13*12*11*10*9!)/(5!*9!)

14 C 5 = (14*13*12*11*10)/(5!)

14 C 5 = (14*13*12*11*10)/(5*4*3*2*1)

14 C 5 = (240240)/(120)

14 C 5 = 2002