A bag has contains 23 balls in which 7 are identical . Then number of ways of selecting 12 balls from bag.

Explanation

Here, `n=23,p=7,r=12 (rgtp)`
`therefore` Required number of selections `=underset(r=5)overset(12)(sum).^(16)C_(r)`
`=.^(16)C_(5)+.^(16)C_(6)+.^(16)C_(7)+.^(16)C_(8)+.^(16)C_(9)+.^(16)C_(10)+.^(16)C_(11)+.^(16)C_(12)`
`=(.^(16)C_(5)+.^(16)C_(6))+(.^(16)C_(7)+.^(16)C_(8))+(.^(16)C_(9)+.^(16)C_(10))+(.^(16)C_(11)+.^(16)C_(12))`
`=.^(17)C_(6)+.^(17)C_(8)+.^(17)C_(10)+.^(17)C_(12)" "[because .^(n)C_(r)+.^(n)C_(r-1)=.^(n+1)C_(r)]`
`=.^(17)C_(11)+.^(17)C_(9)+.^(17)C_(10)+.^(17)C_(12)" "[because.^(n)C_(r)=.^(n)C_(n-r)]`
`=(.^(17)C_(11)+.^(17)C_(12))+(.^(17)C_(9)+^(17)C_(10))`
`=.^(18)C_(12)+.^(18)C_(10)=.^(18)C_(6)+.^(18)C_(8)`