The word SUMMERTIME consists of the letters: S, U, M, M, E, R, T, I, M, E.

We need to select 4 letters including at least one M and exactly one E.

Consider the scenarios:

1. Scenario 1: EMMM

We have already selected 3 M's and 1 E. The number of ways to choose 0 more letters from the remaining 5 letters (S, U, R, T, I) is:

\(\binom{5}{0} = 1\)

2. Scenario 2: EMM_

We have selected 2 M's and 1 E. We need to choose 1 more letter from the remaining 5 letters (S, U, R, T, I):

\(\binom{5}{1} = 5\)

3. Scenario 3: EM__

We have selected 1 M and 1 E. We need to choose 2 more letters from the remaining 5 letters (S, U, R, T, I):

\(\binom{5}{2} = 10\)

Summing the number of ways for the scenarios gives:

\(1 + 5 + 10 = 16\)