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 8 letters. But the letter 'e' appears 3 times. The formula for permutations of a word with repeated letters is $\frac{n!}{n_1!}$, where $n$ is the total number of letters and $n_1$ is the number of times a repeated - letter appears. Here, $n = 8$ and $n_1=3$. So the number of arrangements is $\frac{8!}{3!}=\frac{8\times7\times6\times5\times4\times3!}{3!}=6720$.

Step2: Calculate probability

There is only 1 correct arrangement (the word "generally" itself). The probability $P$ of getting the correct arrangement out of all possible arrangements is the number of favorable outcomes (1) divided by the number of total outcomes. So $P=\frac{1}{6720}$.

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 $\frac{1}{6720}$.