Two letters are chosen randomly from the word ASSESSMENT. Find the probability that either both letters chosen are the same or both letters are vowels.

To solve this problem, we first need to understand the composition of the word ASSESSMENT. It has 10 letters in total: 1 A, 2 S's, 2 E's, 2 S's, 1 M, 1 N, and 1 T.

Step 1: Calculate the total number of ways to choose 2 letters from 10. This can be done using the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items, and r is the number of items to choose. So, C(10, 2) = 10! / [2!(10-2)!] = 45.

Step 2: Calculate the number of ways to choose 2 same letters. We have 3 S's and 2 E's, so we can choose 2 same letters in C(3, 2) + C(2, 2) = 3 + 1 = 4 ways.

Step 3: Calculate the number of ways to choose 2 vowels. We have 1 A, 2 E's, so we can choose 2 vowels in C(3, 2) = 3 ways.

Step 4: Add the number of ways to choose 2 same letters and the number of ways to choose 2 vowels. But notice that choosing 2 E's has been counted twice, so we subtract the number of ways to choose 2 E's. So, the total number of favorable ways = 4 + 3 - 1 = 6.

Step 5: Calculate the probability. The probability is the number of favorable ways divided by the total number of ways. So, the probability = 6 / 45 = 2 / 15.