Question

A club with 15 women and 12 men need to form a committee that consists of a president, a vice president, a secretary, and a treasurer. how many committees are possible…

a. if the committee must have two women and two men?

Answer 1

The selection of r objects out of n, can be done in C(n, r) many ways,

where
`\displaystyle{ C(n, r) = (n!)/(r!(n-r)!)`, r! being
`1\cdot2\cdot...\cdot r`.


Thus, 2 women out of 15 can be selected in a total of C(15, 2) many ways, and 2 men out of 12, can be selected in C(12, 2) many ways.

Any possible pair of women can be combined with any pair of men, thus there are a total of
`C(15, 2)\cdot C(12, 2)` many ways of forming the committee.


`C(15, 2)\cdot C(12, 2)= \displaystyle{ (15!)/(2!13!)\cdot (12!)/(2!10!)= \displaystyle{ (15\cdot14\cdot13!)/(2!13!)\cdot (12\cdot11\cdot10!)/(2!10!)`

`\displaystyle{ = (15\cdot14)/(2)\cdot (12\cdot11)/(2)=15\cdot 7\cdot6\cdot11= 6,930`

Answer: 6,930