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:
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When counting arrangements with restrictions, first consider how the restrictions limit the choices (e.g., repeating letters), and then use combinations and permutations to calculate the total number of valid arrangements.
Step 1: The word "MATHS" contains 5 distinct letters: M, A, T, H, and S. We are required to form 6-letter words where each letter that appears must appear at least twice. Step 2: The only way to satisfy the condition of having each letter that appears at least twice in a 6-letter word is by using exactly 2 of each of 2 letters. Step 3: The number of ways to choose 2 letters from the 5 available letters is \( \binom{5}{2} \), and for each choice of letters, the 6 positions can be arranged in \( \frac{6!}{2!2!} \) ways. Thus, the total number of such 6-letter words is calculated.
