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

Step 1: Analyze the constraints

We need to form 6-letter words using the letters M, A, T, H, S. The crucial constraint is that any letter chosen to be part of the 6-letter word must appear at least twice. This means we cannot use a letter only once.

Step 2: Identify possible letter combinations

Given the constraint that each letter must appear at least twice, and the word length is 6, we can identify three possible patterns for the letters in the word. These patterns cover all ways to sum to 6 using numbers that are at least 2.

Step 3: Calculate words for Case 1 (AABBCC)

First, we choose 3 distinct letters from the 5 available letters (M, A, T, H, S). Then, for each set of 3 chosen letters, say A, B, C, we form a 6-letter word where A appears twice, B appears twice, and C appears twice. The number of permutations for such a word is given by the multinomial coefficient formula $\frac{n!}{n_1!n_2!...n_k!}$.

Step 4: Calculate words for Case 2 (AAAABB)

For this case, we first choose one letter that will appear four times. Then, from the remaining four letters, we choose one letter that will appear twice. Finally, we calculate the number of distinct permutations for these chosen letters using the multinomial coefficient formula.

Step 5: Calculate words for Case 3 (AAAAAA)

Here, we choose one letter from the five available letters, and this letter will be repeated six times to form the 6-letter word. Since all letters are identical, there is only one way to arrange them for each choice of letter.

Step 6: Sum up the results from all cases

To find the total number of possible 6-letter words, we sum the number of words calculated for each of the three distinct cases identified. These cases are mutually exclusive and exhaustive, covering all possibilities under the given constraints.

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