Question

A committee of $5$ is to be formed out of $6$ gents and $4$ ladies. in how many ways this can be done, if at most two ladies are included?

Answer 1

A committee of 5 persons, consisting of at most two ladies, can be done in the following ways:
(a)Selecting $5$ gents only out of $6.$ This can be done in ${}^6{C}_5$ ways.
(b)Selecting $4$ gents only out of $6$ and $1$ lady out of $4.$ This can be done in ${}^6{C}_4\times 4C_1$ , ways.
(c)Selecting $3$ gents only out of $6$ and $2$ ladies out of $4.$ This can be done in ${}^6{C}_3{\times }^4{C}_2$ ways.
Since the committee is formed in each case, so, the total number of ways of forming the committee
${=}^6{C}_5{+}^6{C}_4{\times }^4{C}_1{+}^4{C}_2=6+60+120=186$