How many arrangements can be made by the letters of word DEFINITION if

Solution:

The word DEFINITION has 10! / (3!2!) = 302,400 different permutations of the letters if there are no restrictions on where the certain letter or letters should be. If we consider all the letters (other than I) to be the same, the three I’s can be placed 10C3 = (10 x 9 x 8) / 3! = 120 ways. However, if the three I’s have to be together, then they only can have 8 ways (notice that we are considering all the other letters as the letter X):

IIIXXXXXXX, XIIIXXXXXX, XXIIIXXXXX, XXXIIIXXXX, XXXXIIIXXX, XXXXXIIIXX, XXXXXXIIIX, XXXXXXXIII

In other words, 8/120, or 1/15, of the 302,400 different permutations have the three I’s together. Thus, the number of permutations of the letters of word DEFINITION where the three I’s are together is 1/15 x 302,400 = 20,160.

Alternate Solution:

Since the three I’s must be together, let’s consider them as one “entity” letter; i.e. there are only 8 letters, which are [III] - N - N - D - E - F - T - O. Using the permutation with indistinguishable items formula, we see that there are 8! / 2! = 20,160 arrangements.

Answer: A