In how many ways can the letters of the word MANAGEMENT be rearranged so that the two As do not appear together?

The word MANAGEMENT is a 10 letter word.
Normally, any 10 letter word can be rearranged in 10! ways.
However, as there are certain letters of the word repeating, we need to account for those. In this case, the letters A, M, E and N repeat twice each.
Therefore, the number of ways in which the letters of the word MANAGEMENT can be rearranged reduces to:
$$\frac{{10!}}{{2! \times 2! \times 2! \times 2!}}$$

The problem requires us to find out the number of outcomes in which the two As do not appear together.

The number of outcomes in which the two As appear together can be found out by considering the two As as one single letter.
Therefore, there will now be only 9 letters of which three of them E, N and M repeat twice. So these 9 letters with 3 of them repeating twice can be rearranged in:
$$\frac{{9!}}{{2! \times 2! \times 2!}}$$   ways.

Therefore, the required answer in which the two As do not appear next to each other