The word 'REMEMBRANCE' consists of the letters: R, E, M, E, M, B, R, A, N, C, E.

Removing M and R, we have the letters: E, E, B, A, N, C, E.

We need to choose 4 letters with at least 2 Es.

Case 1: Exactly 2 Es. Choose 2 more letters from B, A, N, C.

The number of ways to choose 2 Es from 3 Es is \(\binom{3}{2} = 3\).

The number of ways to choose 2 more letters from B, A, N, C is \(\binom{4}{2} = 6\).

Total for this case: \(3 \times 6 = 18\) ways.

Case 2: Exactly 3 Es. Choose 1 more letter from B, A, N, C.

The number of ways to choose 3 Es from 3 Es is \(\binom{3}{3} = 1\).

The number of ways to choose 1 more letter from B, A, N, C is \(\binom{4}{1} = 4\).

Total for this case: \(1 \times 4 = 4\) ways.

Adding both cases: \(18 + 4 = 22\) ways.

Alternatively, choose 2 letters from E, E, E, B, A, N, C (5 letters): \(\binom{5}{2} = 10\) ways.

Thus, the total number of ways is 10.