Mathematics - Probability Question with Solution | TestHub

The required probability $=1$ - probability of each receiving at least one $=1-\frac{n(E)}{n(S)}$.

Now, the number of integral solutions of $x_1+x_2+x_3+x_4+x_5+x_6=10$

Such that $x_1\ge 1,x_2\ge 1,\dots .,x_6\ge 1$ gives $n(E)$ and the number of integral solutions of $x_1+x_2+x_3+x_4+x_5+x_6=10$ such that $x_1\ge 0,x_2\ge 0,\dots ,x_6\ge 0$ gives $n(S)$

∴ The required probability $=1-\frac{{}^{10-1}{C}_{6-1}}{{}^{10+6-1}{C}_{6-1}}=1-\frac{{}^9{C}_5}{{}^{15}{C}_5}=\frac{137}{143}$.