Question

If $4$ dice are rolled once, the number of ways of getting the sum as $10$ is $K$ , then the value of $\frac{K}{10}$ is equal to

Answer 1

$x_{1}+x_{2}+x_{3}+x_{4}=10;$ $1\leq x_{i}\leq 6$
The desired number is the coefficient of $x^{10}$ in
$\left(x + x^{2} + \ldots + x^{6}\right)^{4}=x^{4}\left(1 + x + \ldots + x^{5}\right)^{4}$
or the coefficient of $x^{6}$ in $\left(1 + x + x^{2} + \ldots + x^{5}\right)^{4}$
$=\left(\frac{1 - x^{6}}{1 - x}\right)^{4}=\left(1 - x^{6}\right)^{4}\left(1 - x\right)^{- 4}$
$=\left(1 - 4 x^{6} + \ldots \right)\left(1 - x\right)^{- 4}$
or the coefficient of $x^{6}$ in $\left(1 - x\right)^{- 4}-4\times $ coefficient of $x^{0}$ in $\left(1 - x\right)^{- 4}$
$=^{6 + 4 - 1}C_{4 - 1}-4\times ^{0 + 4 - 1}C_{4 - 1}$
$=^{9}C_{3}-4\times 1=\frac{9 \times 8 \times 7}{3 \times 2}-4=84-4=80$
$\Rightarrow K=80\Rightarrow \frac{K}{10}=8$