Discussion
The 10 letters of the word STATISTICS are arranged randomly. The probability that the 10-letter arrangement starts with CAT is
Answer: C
The probability is \(\frac{1}{240}\).
STATISTICS has S three times, T three times, I twice, and A and C once each.
\[ \begin{aligned} \text{total} &= \frac{10!}{3!\,3!\,2!} \\ &= \frac{3628800}{72} = 50400 \end{aligned} \]
If the arrangement starts with C, A, T, the remaining 7 letters are S, S, S, T, T, I, I:
\[ \begin{aligned} \text{favourable} &= \frac{7!}{3!\,2!\,2!} = 210 \\ P &= \frac{210}{50400} = \frac{1}{240} \end{aligned} \]
Dividing 50400 by 210 gives exactly 240. Counting the remaining 7 letters as if all were different (7! = 5040) would give \(\frac{1}{10}\), far from every option, which is why the repeated S, T and I must be divided out.
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