### From a group of $8$ men and $7$ women, 5 persons are to be selected to form a committee so that at least $4$ men are there on the committee. In how many ways can it be done?

$546$

Step by Step Explanation:
1. We are required to select a $5$-member committe from a group of $8$ men and $7$ women with at least $4$ men. This can be done in $2$ different ways:-
1. $4$ men and $1$ woman
2. $5$ men
2. Number of ways of selecting $4$ men and $1$ woman $= 8 C _4 \times 7 C _1$ $= \dfrac{ 8! }{ 4!(8 - 4)! } \times \dfrac{ 7! }{ 1!(7 - 1)! } = 490$
Number of ways of selecting $5$ men $= 8 C _5$ $= \dfrac{ 8! }{ 5!(8 - 5)! } = 56$
3. Now, required number of ways of selecting a $5$-member committee $= 490 + 56 = 546$
Hence, there are $546$ ways of selecting a $5$-member committee with at least $4$ men.