four dice are thrown. What is the probability that the sum of the numbers appearing on the dice is 23?

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{The} \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{sample} \mathrm{points} \mathrm{are} \\ \mathrm{n}\left(\mathrm{S}\right)= 6^4=1296 \\ \mathrm{Let} \mathrm{E} \mathrm{be} \mathrm{the} \mathrm{event} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{numbers} \mathrm{on} \mathrm{die} \mathrm{is} 23. \\ \mathrm{We} \mathrm{get} \mathrm{a} \mathrm{sum} 23 \mathrm{when} \mathrm{three} 6\text{'}\mathrm{s} \mathrm{appear} \mathrm{and} \mathrm{one} 5 \mathrm{appear}. \\ \mathrm{This} \mathrm{implies} \mathrm{that}, \\ \mathrm{E}=\left\{ \left(6,6,6,5\right), \left(6,6,5,6\right), \left(6,5,6,6\right), \left(5,6,6,6\right)\right\} \\ \mathrm{Therefore} \mathrm{the} \mathrm{required} \mathrm{probability} \mathrm{is}, \\ \mathrm{P}\left(\mathrm{E}\right)=\frac{\mathrm{n}\left(\mathrm{E}\right)}{\mathrm{n}\left(\mathrm{S}\right)} \\ =\frac{4}{1296} \\ =\frac{\mathbf{1}}{\mathbf{324}} \end{gathered}$$

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