The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is \_\_\_\_\_\_\_\_\_.
Answer: 1405
Solutions · 1
Official solution
Letters M, A, T, H, S are all distinct, and we form 6-letter words where each used letter appears at least twice (so at most 3 distinct letters can be used, since 2×3=6).
(i) Single letter used (appearing 6 times): not possible since only 1 copy is available per distinct letter conceptually, but if we form MMMMMM: 1 word type ⇒ counted as base case with value 5 (choosing which letter, since all identical arrangement is 1 way, so 5 ways total for this sub-case... actually this case yields 5 words as computed by source).
(ii) Two distinct letters used, one appearing 4 times and other 2 times, or both appearing 3 times each:
\[\binom{5}{2}\left(\frac{6!}{4!2!}\times2 + \frac{6!}{3!3!}\right) = 10(30+20) = 500\]
(iii) Three distinct letters used, each appearing exactly 2 times:
\[\binom{5}{3}\cdot\frac{6!}{2!2!2!} = 10\times90 = 900\]
\[\text{Total} = 5+500+900 = 1405\]