Committee of Four

Probability Level 2

A club has 10 members, five male and five female. In how many ways can an executive committee of four members be chosen if at least one has to be male?

(NSML 2007)


The answer is 205.

Chenjia Lin by Chenjia Lin , 20, USA

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2 solutions

Chenjia Lin
Oct 17, 2015

In the committee of four, one person must be male. Which means that the number of males will be greater than 1. There are multiple ways to form a committee, and they are: 

 - 1 male (out of the five) and 3 females (out of the five)
            or
 - 2 males (out of the five) and 2 females (out of the five)
            or
 - 3 males (out of the five) and 1 female (out of the five)
            or
 - 4 males (out of the five) and 0 females (out of the five)

Notice that I used "and" and "or" to describe the number of males and females possible on the committee. When a problem says "and", you multiply. When a problem says "or", you add. Thus, these cases translate to:

(nCr(5,1) x nCr(5,3)) + (nCr(5,2) x nCr(5,2)) + (nCr(5,3) x nCr(5,1)) + (nCr(5,4) x nCr(5,0))

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Oh god!!!!Damn it!I forgot that the order does not matter. Anyway, nice solution.Upvoted!

Anibrata Bhattacharya - 5 years, 7 months ago
Prince Loomba
Jan 22, 2016

Total-no male i.e. 10C4-5C4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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