If four dice are thrown together. Probability that the sum of the number appearing on them is 13, is

$\frac{35}{324}$

The total number of elementary events associated with the experiment of throwing four dice is $6×6×6×6=6^4$.

Favourable number of elementary events

= Coefficient of $x^{13}$ in $(x^1 + x^2 +x^3 +x^4 +x^5+x^6)^4$

= Coefficient of $x^9$ in $(1 + x + x^2 + .......+x^5)^4$

= Coefficient of $x^9$ in $\left(\frac{1-x^6}{1-x}\right)^4$

= Coefficient of $x^9$ in $(1-x^6)^4 (1-x)^{-4}$

= Coefficient of $x^9$ in $(1- {^4C}_1 x^6 + {^4C}_2 x^{12} -....) (1-x)^{-4}$

= Coefficient of $x^9$ in $(1-x)^{-4} - {^4C}_1 × $ Coefficient of $x^3$ in $(1-x)^{-4}$

$= {^{9+4-1}C}_{4-1} - {^4C}_1 × {^{4-1}C}_{4-1}$     [∵ Coeff. of $x^n$ in $(1-x)^{-r} = {^{n+r-1}C}_{r-1}]$

$= {^{12}C}_3 -4 × {^6C}_3= 140$

∴ Required probability $=\frac{140}{6^4}=\frac{35}{324}$