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(a) All Ss are consecutive, AND (b) Vowels appear in alphabetical order.

So clearly there are 10 letters in STATISTICS: 3 Ss, 3 Ts, 2 Is, 1 A, and 1 C. Because the Ss must be consecutive, we can treat the 3 Ss as one letter - 1 way. To have the vowels in alphabetical order we must have AII and we can also make this one letter - 1 way. So now instead of having 10 spaces, we only have 6. Am I right so far? How do I distribute the remaining letters?

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    Treating the $3$ S's as a single letter is fine, but you can't make the vowels into one letter, because they may be spread along the string. – 2017-01-20

1 Answers 1

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Your treatment of the Ss is completely right.

You made a minor error in the treatment of the vowels. Yes you can treat them as one type of letter, call it $\nu$, but there are still three instances of $\nu$ in the arangments.

So the problem is equivalent to the number of arrangements of $8$ symbols, two of which (SSS and C) are distinct from all other symbols, and the other six of which consist of two groups of three identical symbols ($\nu\nu\nu$ and TTT).

You then can reason:

  • $8$ places to put the SSS.

  • Then $7$ slots to put the C.

  • Then $\binom{6}{3} = 20$ ways to choose the three slots to put the Ts.

  • At which point, the slots to put the vowels are fully determined, and there is only one arrangement of the vowels allowed among those slots.

The answer, then, is

$$ 8\cdot 7\cdot 20 = 1120$$