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 1/6720 (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: Count total letters and duplicates

The word "generally" has 8 letters. The letter 'e' appears 3 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 \(n = 8\) and \(n_1=3\) (for 'e'). So the number of arrangements is \(\frac{8!}{3!}=\frac{8\times7\times6\times5\times4\times3!}{3!}=6720\).

Step3: Calculate probability

There is only 1 correct arrangement (the word "generally" itself). The probability \(P\) of getting the word "generally" in a random arrangement is \(\frac{1}{6720}\).

Answer:

The number of different ways that the letters of "generally" can be arranged is \(6720\).
The probability that the random arrangement of letters will result in "generally" is \(\frac{1}{6720}\).