Question

The number of ways in which $16$ identical things can be distributed among $4$ persons if each person gets at least $3$ things is

• A. $33$
• B. $35$
• C. $38$
• D. None of these

Answer 1

Answer: (B)

$35$

Required number

= coefficient of $x^{16}$ in ${(x^3+x^4+x^5+..+x^7)}^4$

= coefficient of $x^{16}$ in $x^{12}(1+x+...+x^4)$

= coefficient of $x^4$ in ${\left(\frac{1-x^5}{1-x}\right)}^4$

=coefficient of $x^4$ in ${(1-x^5)}^4{(1-x)}^{-4}$

=coefficient of $x^4$ in $(1-4x^5+6x^{10}-...){(1-x)}^{-4}$

coefficient of $x^4$ in ${(1-x)}^{-4}$$={{}^{7}C}_4=35$