Question

how many different ways can the letters of \embarrass\ be arranged? if the letters of \embarrass\ are arranged in a random order, what is the probability that the result will be \embarrass\? the number of different ways that the letters of \embarrass\ can be arranged is (type an integer or a simplified fraction.)

Explanation:

Step1: Count total letters and repeated letters

The word "embarrass" has 9 letters. The letter 'r' appears 2 times and the letter 's' appears 2 times.

Step2: Use permutation formula for multi - sets

The formula for permutations of a multi - set with \(n\) objects where \(n_1\) of one kind, \(n_2\) of another kind, \(\cdots\), \(n_k\) of the \(k\)th kind is \(\frac{n!}{n_1!n_2!\cdots n_k!}\). Here \(n = 9\), \(n_1=2\) (for 'r') and \(n_2 = 2\) (for 's'). So the number of arrangements is \(\frac{9!}{2!2!}\).

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Answer:

90720