Find the number of permutations of letters of the word EXERCISES in which vowels are together?

Treat the vowels as one group, so pretend they are a single letter. That means there are six "letters" to permute. This can be done in 6!/2! =360 ways (divide by 2 because of the repeated "s"). Now, the vowels can also be permuted within their group. This can be done in 4!/3! = 4 ways. The total is 360 *4 = 1440.