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 

(i) atleast two ladies are included. 

(ii) 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	4C2 x 6C3 = \(\frac{4\times3}{2\times1}\times{\frac{6\times5\times4}{3\times2\times1}}\) = 120
	1	4	4C1 x 6C4 = 4 x 6C2 = 4 x \(\frac{6\times5}{2\times1}\) = 60
	0	5	4C0 x 6C5 = 1 x 6C1 = 1 x 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	4C2 x 6C3  = \(\frac{4\times3}{2\times1}\times{\frac{6\times5\times4}{3\times2\times1}}\) = 120
2	3	2	4C3 x 6C2 = \({\frac{4\times3\times2}{3\times2\times1}}\times{\frac{6\times5}{2\times1}}\) = 60
3	4	1	4C4 x 6C1 = 1 x 6 = 6
			Total number of ways = 186