A Committee of 5 persons is to be formed from a group of 6 gentlemen and 4 ladies. In how many ways can this be done if the committee is to be included atleast one lady?

A Committee of 5 persons is to be formed from 6 gentlemen and 4 ladies by taking. 

(i) 1 lady out of 4 and 4 gentlemen out of 6 

(ii) 2 ladies out of 4 and 3 gentlemen out of 6 

(iii) 3 ladies out of 4 and 2 gentlemen out of 6 

(iv) 4 ladies out of 4 and 1 gentlemen out of 6 

In case I the number of ways =  $C14\times C46$ = 4 x 15 = 60 

In case II the number of ways =  $C24\times C36$ = 6 x 20 = 120 

In case III the number of ways =  $C34\times C26$ = 4 x 15 = 60

In case IV the number of ways =  $C44\times C16$ = 1 x 6 = 6 

Hence, the required number of ways = 60 + 120 + 60 + 6 = 246