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

Answer: 8

$x_1+x_2+x_3+x_4=10;$ $1\le x_i\le 6$
The desired number is the coefficient of $x^{10}$ in
${\left(x+x^2+\dots +x^6\right)}^4=x^4{\left(1+x+\dots +x^5\right)}^4$
or the coefficient of $x^6$ in ${\left(1+x+x^2+\dots +x^5\right)}^4$
$={\left(\frac{1-x^6}{1-x}\right)}^4={\left(1-x^6\right)}^4{\left(1-x\right)}^{-4}$
$=\left(1-4x^6+\dots \right){\left(1-x\right)}^{-4}$
or the coefficient of $x^6$ in ${\left(1-x\right)}^{-4}-4\times$ coefficient of $x^0$ in ${\left(1-x\right)}^{-4}$
${=}^{6+4-1}{C}_{4-1}-4{\times }^{0+4-1}{C}_{4-1}$
${=}^9{C}_3-4\times 1=\frac{9\times 8\times 7}{3\times 2}-4=84-4=80$
$\Rightarrow K=80\Rightarrow \frac{K}{10}=8$