Find the probability that in a random arrangement of the letters of the word MATHEMATICS, the two A's are not together.

Step 1: Calculate total number of arrangements

First, we need to find the total number of distinct arrangements of the letters in the word MATHEMATICS. The word has 11 letters in total. We identify the letters that repeat: 'M' appears 2 times, 'A' appears 2 times, and 'T' appears 2 times. The formula for permutations with repetitions is n! / ($n_1! n_2!$ ... $n_k!$), where n is the total number of letters and $n_i$ is the frequency of each repeated letter.

Step 2: Calculate arrangements where the two A's are together

Next, we calculate the number of arrangements where the two 'A's are always together. We can treat the pair 'AA' as a single block. Now, we have 10 units to arrange (M, A, T, H, E, M, A, T, I, C, S becomes M, (AA), T, H, E, M, T, I, C, S). The repeated letters are 'M' (2 times) and 'T' (2 times).

Step 3: Calculate arrangements where the two A's are not together

The number of arrangements where the two 'A's are not together is found by subtracting the number of arrangements where they are together from the total number of arrangements. This is a common strategy for 'not together' problems in combinatorics.

Step 4: Calculate the probability

Finally, we calculate the probability by dividing the number of arrangements where the 'A's are not together by the total number of arrangements. We can simplify the expression by factoring out common terms and performing the division.

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