Question

A committee of 10 persons is to be formed from a group of 10 women and 8 men how many possible committes will have at least 5 women? How many possible committee will have men in majority?

Answer 1

Step-by-step explanation:

2 committees will have at least 5 women

only 1 committee will have men in majority

Answer 2

36873 possible committees will have at least 5 women and 6885 possible committees will have men in majority.

Step-by-step explanation:

A committee of 10 persons is to be formed from a group of 10 women and 8 men.

We have to choose at least 5 women.

We can choose 5 women out of 10 women and 5 men out of 8 men = $10_{C}_{5} \times 8_{C}_5 = 252 \times 56 = 14112 \ ways$

We can choose 6 women out of 10 women and 4 men out of 8 men = $10_{C}_6\times 8_{C}_4=210 \times 70 = 14700 \ ways$

We can choose 7 women out of 10 women and 3 men out of 8 men =$10_{C}_7\times 8_{C}_3=120\times 56=6720 \ ways$

We can choose 8 women out of 10 women and 2 men out of 8 men =$10_{C}_8\times 8_{C}_2=45\times 28 = 1260 \ ways$

We can choose 9 women out of 10 women and 1 men out of 8 men =$10_{C}_9\times 8_{C}_1=10\times 8 = 80 \ ways$

We can choose all 10 women = 1 way

Total possible committee  which have at least 5 women = 14112 + 14700 + 6720 + 1260 + 80 + 1 = 36873 ways

Possible committee in which men in majority = $8_{C}_6\times 10_{C}_4 + 8_{C}_7\times 10_{C}_3+8_{C}_8\times 10_{C}_2\\= 28\times 210+8\times 120+1\times 45\\= 5880+960+45\\=6885 \ ways$

Hence, 36873 possible committees will have at least 5 women and 6885 possible committees will have men in majority.