The word REFRIGERATOR consists of the letters: R, E, F, R, I, G, E, R, A, T, O, R.

We need to select four letters with no Rs and exactly two Es.

First, remove all Rs from the word, leaving: E, F, I, G, E, A, T, O.

Now, we have 8 letters: E, E, F, I, G, A, T, O.

We need to select 4 letters, including exactly 2 Es.

\(Choose 2 Es: inom{2}{2} = 1.\)

\(Choose 2 more letters from the remaining 6 letters (F, I, G, A, T, O): inom{6}{2} = 15.\)

\(Thus, the total number of selections is: 1 imes 15 = 15.\)