For the equation `x + y + z +omega= 19,` the number of positive integral solutions is equal to-

Options

A. The number of ways `15` identical things can be distributed among `4` persons.
B. The number of ways `19` identical things can be distributed among `4` persons.
C. Coefficient of `x^(19)` in `(x^(0) + x^(1) + x^(2) + "………" + x^(19))^(4)`
D. Coefficient of `x^(19)` in `(x + x^(2) + x^(3) + "………" + x^(19))^(4)`

Answer

A, D

Explanation

Number of positive integral solution `= .^(19-1)C_(4-1) = .^(18)C_(3)`
`=` Coefficients of `x^(19)` in `(x+x^(2)+x^(3)+"…….."+x^(19))^(4)`
Number of ways in which `15` identical things can be distributed among `4` persons `= .^(15 + 4 -1)C_(15) = .^(18)C_(15) = .^(18)C_(3)`