Question

A coin whose faces are marked $2,3$ is thrown $5$ times $:$ what is the chance of obtaining a total of $12?$

Answer 1

Coin is tossed five times then possible number of outcomes $=2^5=32$

Favourable outcomes $=$ coefficient of $x^{12}$ ins the expansion of ${(x^2+x^3)}^5$

$=$ coefficient of $x^{12}$ in the expansion of $x^{10}{(1+x)}^5$

$=$ coefficient of $x^2$ in the expansion of ${(1+x)}^5$

${=}^5{C}_3{\left(1\right)}^3=10$

$\therefore$ desired probability $=\frac{10}{32}=\frac{5}{16}$