Question

A committee of $8$ persons is to be constituted from a group of $5$ women and $7$ men. In how many ways can this be done? In how many ways can the committee be formed if it consists of at-least $3$ women and $3$ men? What value is depicted by taking men and women in forming the committee?

Answer 1

there are $5$ women and $7$ men

among them we should form committee of $8$ members

therefore the possible outcomes could be

${}^{12}{C}_8=495$

now

given that there must me at least $3$ women and $3$ men

therefore the possible case can be

men women

$3$ $5$

$4$ $4$

$5$ $3$

therefore the total no of cases are ${}^7{C_3}^5{C}_5{+}^7{C_4}^5{C}_4{+}^7{C_5}^5{C}_3=420$