Question

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

Explanation:

Step1: Calculate total arrangements

The word "generally" has 9 letters. But the letter 'e' appears 3 times. The formula for permutations of a multi - set 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 \(n = 9\) and \(p_1=3\) (for 'e'). So the number of arrangements is \(\frac{9!}{3!}=\frac{9\times8\times7\times6\times5\times4\times3!}{3!}=60480\).

Step2: Calculate probability

There is only 1 correct arrangement (the word "generally" itself). Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So the probability is \(\frac{1}{60480}\).

Answer:

The number of different ways that the letters of "generally" can be arranged is \(60480\).
The probability that the random arrangement of letters will result in "generally" is \(\frac{1}{60480}\).