Question

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

Answer 1

Number of women = 10
Number of men = 8
Number of persons in the team = 10
A committee of 10 persons consisting of at least 5 women can be formed as follows:
(I) 5 women and 5 men or
(II) 6 women and 4 men or
(III) 7 women and 3 man or
(IV) 8 women and 2 man or
(V) 9 women and 1 man or
(VI) 10 women
 The number of ways of forming the committee:
(I) 5 women and 5 men

= `""^10"C"_5 xx ""^8"C"_5`

= `(10 xx 9 xx 8 xx 7 xx 6 xx 5 xx 4)/(5 xx 4 xx 3 xx 2 xx 1) xx (8 xx 7 xx 6)/(3 xx 2 xx 1)`

= (2 × 9 × 2 × 7) × (8 × 7)
= 14112
(II) 6 women and 4 men

= `""^10"C"_6 xx ""^8"C"_4`

= `(10 xx 9 xx 8 xx 7)/(4 xx 3 xx 2 xx 1) xx (8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1)`

= (5 × 2 × 3 × 7) × (2 × 7 × 5)
= 14700
(III) 7 women and 3 men

= `""^10"C"_7 xx ""^8"C"_3`

= `(10 xx 9 xx 8)/(3 xx 2 xx 1) xx (8 xx 7 xx 6)/(3 xx 2 xx 1)`

= (10 × 12) × (8 × 7)
= 6720
(IV) 8 women and 2 men

= `""^10"C"_8 xx ""^8"C"_2`

= `(10 xx 9)/(2 xx 1) xx (8 xx 7)/(2 xx 1)`

= (5 × 9) × (4 × 7)
= 1260
(V) 9 women and 1 men

= `""^10"C"_9 xx ""^8"C"_1`

= `10/1 xx 8/1`

= 80
(VI) 10 women

= `""^10"C"_10`
= 1
Hence, the number of ways of forming the required committee
= 14112 + 14700 + 6720 + 1260 + 80 + 1
= 36873
For men to be in majority, the committee should have 6 or more men.
Following are the possibilities:
(I) 6 men and 4 women or
(II) 7 men and 3 women or
(III) 8 men and 2 women
The number of ways of forming the committee:
(I) 6 men and 4 women

= `""^8"C"_6 xx  ""^10"C"_4`

= `(8 xx 7)/(1 xx 2) xx (10 xx9 xx 8 xx 7)/(1 xx 2 xx 3 xx 4)`

= 5880
(II) 7 men and 3 Women

= `""^8"C"_7 xx  ""^10"C"_3`

= `8 xx (10 xx 9 xx 8)/(1 xx 2 xx 3)`

= 960
(III) 8 men and 2 women

= `""^8"C"_8 xx  ""^10"C"_2`

= `1 xx (10 xx 9)/(1 xx 2)`

= 45
Hence, number of ways of forming the required committee
= 5880 + 960 + 45
= 6885