The word TOADSTOOL consists of the letters: T, O, A, D, S, T, O, O, L.

We need to select 5 letters including at least 2 Os and at least 1 T.

Consider the following cases:

1. Case 1: 2 Os and 1 T (OOT__). We need 2 more letters from A, D, S, L.

   The number of ways to choose 2 more letters from 4 is given by: \(\binom{4}{2} = 6\).

2. Case 2: 2 Os and 2 Ts (OOTT_). We need 1 more letter from A, D, S, L.

   The number of ways to choose 1 more letter from 4 is given by: \(\binom{4}{1} = 4\).

3. Case 3: 3 Os and 1 T (OOOT_). We need 1 more letter from A, D, S, L.

   The number of ways to choose 1 more letter from 4 is given by: \(\binom{4}{1} = 4\).

4. Case 4: 3 Os and 2 Ts (OOOTT). No more letters needed.

   There is only 1 way to choose this combination.

Adding all the cases together gives: \(6 + 4 + 4 + 1 = 15\).