The number of ways of getting a sum 16 on throwing a dice four times is______

Question

Aicharya · Accepted Answer

$\begin{aligned} & \left(x^1+x^2 \ldots+x^6ight)^4 \ & x^4 \cdot\left(\frac{1-x^6}{1-x}ight)^4 \ & x^4 \cdot\left(1-x^6ight)^4 \cdot(1-x)^{-4} \ & x^4\left[1-4 x^6+6 x^{12} \ldots .ight]\left[(1-x)^{-4}ight] \ & \left(x^4-4 x^{10}+6 x^{16} \ldotsight)(1-x)^{-4} \ & \left(x^4-4 x^{10}+6 x^{16}ight)\left(1+{ }^{15} C_{12} x^{12}+{ }^9 C_6 x^6 \ldots .ight) \ & \left({ }^{15} C_{12}-4 \cdot{ }^9 C_6+6ight) x^{16} \ & \left({ }^{15} C_3-4 \cdot{ }^9 C_6+6ight) \ & =35 	imes 13-6 	imes 8 	imes 7+6 \ & =455-336+6 \ & =125\end{aligned}$

The number of ways of getting a sum 16 on throwing a dice four times is______

Solution