Question

The number of selections of four letters from the letters of the word ASSASSINATION is:

• A. 72
• B. 71
• C. 66
• D. 52

Hint:

For selections with repetition, break the problem into cases based on letter frequencies and sum the valid cases.

Answer 1

The Correct Option is A

The word ASSASSINATION consists of 12 letters, with frequencies:
- A → 3 times
- S → 4 times
- I → 2 times
- N → 2 times
- T → 1 time
- O → 1 time
To select 4 letters, we consider different cases:
1. All 4 different letters: \( \binom{6}{4} = 15 \)
2. Two same, two different: \( \binom{4}{1} \times \binom{5}{2} = 40 \)
3. Two pairs of same letters: \( \binom{4}{2} = 6 \)
4. Three same, one different: \( \binom{2}{1} \times \binom{5}{1} = 10 \)
5. All four same (only S has 4 occurrences):** \( 1 \)
Total:
\[ 15 + 40 + 6 + 10 + 1 = 72 \]
Thus, the correct answer is 72.