Question

how many different ways can the letters of \personnel\ be arranged? if the letters of \personnel\ are arranged in a random order, what is the probability that the result will be \personnel\?
the number of different ways that the letters of \personnel\ can be arranged is 90720 (type an integer or a simplified fraction.)
the probability that the random arrangement of letters will result in \personnel\ is (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate total arrangements

The word "personnel" has 9 letters, with 3 'n's and 2 'e's. The formula for permutations of a multi - set with \(n\) objects, where there are \(n_1\) 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 \(n = 9\), \(n_1=3\) (for 'n') and \(n_2 = 2\) (for 'e'). So the number of arrangements is \(\frac{9!}{3!2!}=\frac{9\times8\times7\times6\times5\times4\times3!}{3!\times2\times 1}=90720\).

Step2: Calculate probability

The probability \(P\) of getting a specific arrangement out of all possible arrangements is given by \(P=\frac{1}{\text{Total number of arrangements}}\). Since there is only 1 correct arrangement of "personnel" and 90720 total arrangements, the probability is \(\frac{1}{90720}\).

Answer:

The probability that the random arrangement of letters will result in "personnel" is \(\frac{1}{90720}\)