Question

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

Answer 1

A committee of person, consisting of at least 2 ladies, can be formed in the following ways :
(a) Selecting 2 ladies out of 4 and 3 gents out of 6. This can be done in ${}^4{C}_2{\times }^6{C}_3$ ways.
(b) Selecting 3 ladies out of 4 and 2 gents out of6. This can be done in ${}^4{C}_3{\times }^6{C}_2$ ways.
(c) Selecting 4 ladies out of 4 and 1 gents out of 6. This can be done in ${}^4{C}_3{\times }^6{C}_1$ ways.
Since the committee is formed in each case, therefore by the fundamental principle of addition, the total number of ways of forming the committee.
${=}^4{C}_2{\times }^6{C}_3{+}^4{C}_3{\times }^6{C}_2{+}^4{C}_4{\times }^6{C}_1=120+60+6=186$