How many ways can the word great be arranged?

If you mean re-arranging the letters 'g', 'r', 'e', 'a', and 't' in different sequences, then: We have five letters. The first letter can be any one of the five.

For each choice, the second letter can be any one of the remaining four.

For each choice, the third letter can be any one of the remaining three.

For each choice, the fourth letter can be either one of the remaining two.

Then there is only one letter left to put in the fifth place. So there are (5 x 4 x 3 x 2 x 1) = 120 different ways to carry out this process.