Question

a classic counting problem is to determine the number of different ways that the letters of generally can be arranged. find that number. if the letters are mixed up in a random sequence, what is the probability that the letters will be in alphabetical order?
the number of different ways that the letters of generally can be arranged is (square). (type an integer or a simplified fraction.)

Explanation:

Step1: Count the number of letters

The word "generally" has 9 letters. But the letter 'e' appears 3 times and the letter 'l' appears 2 times.

Step2: Use the permutation formula for multi - sets

The number of permutations of a word with \(n\) objects, where \(p_1\) of one kind, \(p_2\) of another kind, \(\cdots\), \(p_k\) of the \(k\)th kind is \(\frac{n!}{p_1!p_2!\cdots p_k!}\). Here \(n = 9\), \(p_1=3\) (for 'e') and \(p_2 = 2\) (for 'l'). So the number of arrangements is \(\frac{9!}{3!2!}=\frac{9\times8\times7\times6\times5\times4\times3!}{3!\times2\times 1}=\frac{9\times8\times7\times6\times5\times4}{2\times1}=30240\).

Answer:

30240