The word REQUIREMENT consists of the letters: R, E, Q, U, I, R, E, M, E, N, T.

We need to select 5 letters containing at least two Es and at least one R.

Consider the following cases:

1. Case 1: 2 Es and 1 R

We have E, E, R, and need to choose 2 more letters from the remaining 8 letters (Q, U, I, M, N, T, R, E).

The number of ways to choose 2 more letters is given by: \(\binom{8}{2} = 28\).

2. Case 2: 3 Es and 1 R

We have E, E, E, R, and need to choose 1 more letter from the remaining 7 letters (Q, U, I, M, N, T, R).

The number of ways to choose 1 more letter is given by: \(\binom{7}{1} = 7\).

3. Case 3: 2 Es and 2 Rs

We have E, E, R, R, and need to choose 1 more letter from the remaining 7 letters (Q, U, I, M, N, T, E).

The number of ways to choose 1 more letter is given by: \(\binom{7}{1} = 7\).

4. Case 4: 3 Es and 2 Rs

We have E, E, E, R, R, and do not need to choose any more letters.

There is only 1 way to choose 0 more letters: \(\binom{6}{0} = 1\).

Adding all the cases together, we get:

\(15 + 6 + 6 + 1 = 28\).

Thus, the total number of possible selections is 28.