The word STEEPLECHASE consists of the letters: S, T, E, E, P, L, E, C, H, A, S, E.

We need to select four letters including exactly one S. There are two S's in the word, so we choose one S and three other letters from the remaining 10 letters.

First, count the distinct letters after choosing one S: T, E, P, L, C, H, A, E, E, E.

We can choose the remaining three letters in the following ways:

1. Choose 3 E's: inom{6}{1} = 6
2. Choose 2 E's and 1 other letter: inom{6}{2} = 15
3. Choose 1 E and 2 other letters: inom{6}{3} = 20

Adding these scenarios gives the total number of selections:

\(6 + 15 + 20 = 41\)

Therefore, the total number of different selections is 42.