2.
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2. Forming strings The letters A, A, B, M, and T can be used to form 5-letter strings such as MABAT or MAABT. You have to use all these letters to form 5-letter strings such that two occurrences of the letter A are separated by at least one letter. In the given context, how many strings can be formed?
Show answer & explanation
The total number of permutations of the letters A, A, B, M, T is calculated, and the number of permutations where the two A's are adjacent is subtracted. This uses the complement method to find the number of strings where A's are separated by at least one letter.
Step-by-step Derivation:
Step 1: Calculate the total number of permutations of the letters {A, A, B, M, T}. Since there are 5 letters and A is repeated twice, the total permutations are 5! / 2! = 120 / 2 = 60.
Step 2: Calculate the number of permutations where the two A's are together (adjacent). Treat the two A's as a single block (AA). Now we have 4 entities to arrange: (AA), B, M, T. The number of permutations is 4! = 24.
Step 3: Subtract the cases where A's are together from the total permutations to find the cases where A's are separated. 60 - 24 = 36.