Out of 5 identical apples, 10 identical mangoes and 15 identical oranges, the number of ways of distributing 15 fruits each to two persons, is

$x_1+x_2+x_3=15$

$0\le x_1\le 5,0\le x_2\le 10,0\le x_3\le 15$

Consider the coefficient of $x^{15}$ in $\frac{\left(1-x^6\right)\left(1-x^{11}\right)\left(1-x^{16}\right)}{{\left(1-x\right)}^3}$

The above expression can be written as $\left(1-x^6-x^{11}\right)\left(1-x^{16}\right){\left(1-x\right)}^{-3}=\left(1-x^6-x^{11}\right){\left(1-x\right)}^{-3}$

Coefficient of $x^{15}$ in the above expression is $\begin{array}{rcl}{\left(3+15-1\right)}_{C_{15}}-{\left(3+9-1\right)}_{C_9}-{\left(3+4-1\right)}_{C_4}& =& {17}_{C_{15}}-{11}_{C_9}-6_{C_4}\\ & =& \frac{17·16}{2}-\frac{11·10}{2}-\frac{6·5}{2}\\ & =& 132-55-15\\ & =& 66\end{array}$