The number of ways of distributing 15 apples to three persons $\mathrm{A}, \mathrm{B}, \mathrm{C}$ such that…
The number of ways of distributing 15 apples to three persons $\mathrm{A}, \mathrm{B}, \mathrm{C}$ such that A and C each get at least 2 apples and B gets at most 5 apples is
57
131
156
251
Solution
$\begin{aligned} & \text { Required number of ways } \\ & \text { = Coeff of } x^{15} \text { in } \\ & \left(x^2+x^3+x^4+\ldots\right)^2\left(1+x+x^2+x^3+x^4+x^5\right) \\ & =\text { Coeff of } x^{11} \text { in }(1-x)^{-2}\left(1-x^6\right)(1-x)^{-1}-\text { Coeff of } \\ & x^{11} \text { in }(1-x)^{-3}\left(1-x^6\right) \\ & ={ }^{11+3-1} C_{11}-{ }^{5+3-1} C_5={ }^{13} C_{11}-{ }^7 C_5=57\end{aligned}$