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

1. atleast two ladies are included.
2. atmost two ladies are included.

Answer 1

(i) A committee of 5 is to be formed.

Possibilities	Ladies (4)	Gents (6)	Combinations
1	2	3	`4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120
	1	4	`4"C"_1 xx 6"C"_4 = 4 xx 6"C"_2 = 4 xx (6xx5)/(2xx1) = 60`
	0	5	4C0 × 6C5 = 1 × 6C1 = 1 × 6 = 6
			Total number of ways = 186

(ii) Almost two ladies are included means maximum of two ladies are included.

Possibilities	Ladies (4)	Gents (6)	Combinations
1	2	3	`4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120
2	3	2	`4"C"_3 xx 6"C"_2 = (4xx3xx2)/(3xx2xx1) xx (6xx5)/(2xx1)  = 60`
3	4	1	4C4 × 6C1 = 1 × 6 = 6
			Total number of ways = 186