The number of ways in which `20` identical coins be distributed in `4` persons if each person receive at least `2` coins and atmost `5` coins, are

Options

A. `.^(15)C_(12) - 4. .^(11)C_(8) + 6. .^(7)C_(4) - 4`
B. `.^(15)C_(3) - 6. . ^(11)C_(8) + .^(7)C_(4)`
C. `.^(23)C_(3)`
D. `.^(15)C_(0)`

Answer

A, D

Explanation

Number of ways `=` coefficient of `x^(20)` in
`(x^(2) + x^(3) + x^(4) + x^(5))^(4) = x^(B) (1 + x + x^(2) + x^(3))^(4)`
`=` coefficient of `x^(12)` in `((1 - x^(4))/(1 - x))^(4) = (1 - 4^(4))^(4)(1 - x)^(-4)`
`=.^(15)C_(12) - 4 . .^(11)C_(8) + 6. .^(7)(C_(4) - 4 = 1 = .^(15)C_(0)`