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June 2014 p61 q6
2756
4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain
(iii) exactly 1 N and 1 A,
(iv) exactly 1 N?
Solution
(iii) To select exactly 1 N and 1 A, we choose 1 N and 1 A from the letters TANZANIA. The remaining letters are T, Z, I, and the other A. We need to choose 2 more letters from these 3 letters. This can be done in \(\binom{3}{2} = 3\) ways.
(iv) To select exactly 1 N, we consider different cases based on the number of A's:
0 A's: Select N and 3 other letters from T, Z, I, A. This is 1 way: N, T, Z, I.
2 A's: Select N, 2 A's, and 1 other letter from T, Z, I. This can be done in \(\binom{3}{1} = 3\) ways.
3 A's: Select N and 3 A's. This is 1 way: N, A, A, A.
Adding these, the total number of ways is \(1 + 3 + 1 = 8\) ways.
