Question

The number of all possible combinations of 4 letters which are taken from the letters of the word ACCOMMODATION is

• A. \(167\)
• B. \(161\)
• C. \(160\)
• D. \(157\)

Hint:

When selecting letters from a word with repetitions, use combinatorics with case-by-case analysis.

Answer 1

The Correct Option is A

The word ACCOMMODATION has the following letters and their counts: A: 2 C: 2 O: 3 M: 2 D: 1 I: 1 N: 1 T: 1 Total letters: 13 Distinct letters: A, C, O, M, D, I, N, T (8 distinct letters) We want to find the number of combinations of 4 letters. Case 1: All 4 letters are distinct. We have 8 distinct letters, so the number of combinations is \(\binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\) Case 2: 2 letters are same, 2 are distinct. We have 4 letters that repeat (A, C, O, M). We choose 1 of them, which is \(\binom{4}{1}\). Then we choose 2 distinct letters from the remaining 7 distinct letters, which is \(\binom{7}{2} = \frac{7 \times 6}{2} = 21\). So the number of combinations is \(\binom{4}{1} \times \binom{7}{2} = 4 \times 21 = 84\) Case 3: 2 letters are same, 2 letters are same. We choose 2 letters from the 4 letters that repeat (A, C, O, M), which is \(\binom{4}{2} = \frac{4 \times 3}{2} = 6\) Case 4: 3 letters are same, 1 is distinct. Only 'O' appears 3 times. We choose 'O' and 1 distinct letter from the remaining 7 distinct letters. So the number of combinations is \(\binom{1}{1} \times \binom{7}{1} = 1 \times 7 = 7\) Case 5: 4 letters are same. No letter appears 4 times, so this case is not possible. Total number of combinations = 70 + 84 + 6 + 7 = 167 Therefore, the number of all possible combinations of 4 letters is 167. Final Answer: The final answer is $\boxed{(1)}$