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Nov 2019 p62 q7
2744
Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T.
Solution
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:
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:
\(\binom{4}{2} = 6\).
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:
\(\binom{4}{1} = 4\).
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:
\(\binom{4}{1} = 4\).
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:
\(6 + 4 + 4 + 1 = 15\).
