Q.Find the number of distinct arrangements of the letters of the word 'STATISTICS'.

'STATISTICS' has 10 letters but several repeat, so the total arrangements of all 10 letters must be divided by the factorial of each repeated letter's count.

The word has 10 letters with S repeated 3 times, T repeated 3 times, I repeated 2 times, and A, C once each, so the count is $\dfrac{10!}{3!\,3!\,2!}$.

First count the letters of 'STATISTICS': S-T-A-T-I-S-T-I-C-S, which is 10 letters in total. Counting repeats: S appears 3 times, T appears 3 times, I appears 2 times, and A and C appear once each ($3+3+2+1+1=10$, confirming the count).

The number of distinct arrangements of $n$ objects with repeated groups of sizes $p_1,p_2,\ldots$ is $\dfrac{n!}{p_1!\,p_2!\cdots}$. Here:

Check (independent verification): recompute the denominator a different way — since $1!=1$ contributes nothing, the denominator is simply $3!\times3!\times2! = 6\times6\times2=72$, matching what was used above; and $10!=3628800$ can be independently verified as $10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, which multiplies out to $3628800$. Dividing again, $3628800\div72=50400$, the same result.

Miscounting how many times each letter repeats (a very easy slip in a 10-letter word) — always list every letter with its position before counting, rather than counting from memory of how the word 'sounds'.