Using the letters in the word ADDITION, find the number of permutations that can be formed using 3 letters at a time.
A. 300
B. 323
C. 336
D. 345
To figure out the number of permutations of the word ADDITION using 3 letters at a time, we need to consider the fact that there are repeating letters in the word: 'D' appears twice and 'I' appears twice. The word 'ADDITION' consists of 8 letters in total.
When we are taking 3 letters at a time, there are several cases to consider with respect to repeating letters:
Case 1: All 3 letters are different.
There are 6 unique letters in the word (A, D, I, T, O, N). The number of permutations of picking 3 different letters out of 6 is given by 6P3 (6 permute 3), which equals:
6P3 = 6! / (6-3)! = 6! / 3! = (6 × 5 × 4 × 3 × 2 × 1) / (3 × 2 × 1) = 6 × 5 × 4 = 120 permutations
Case 2: Two are the same, and one is different (this happens with 'D' and 'I').
We have 2 repeating letters that can be selected in this manner: D and I.
For 'D':
- Choose 'DD' in 1 way.
- Choose the 1 different letter from the remaining 5 unique letters in 5 ways.
So this gives us 1 × 5 = 5 permutations for D.
Repeat the same process for 'I', resulting in another 5 permutations.
Therefore, for repeating letters, we get 5 (for D) + 5 (for I) = 10 permutations
Adding the permutations from both cases together:
Case 1 + Case 2 = 120 (all different) + 10 (two the same, one different) = 130 permutations
Thus, 130 is the number of permutations that can be formed using 3 letters at a time from the word ADDITION.
The provided answer options do not match the calculated result; it's possible that there may be a mistake in the options or in the interpretation of the question. If you are referring specifically to different permutations disregarding identical outcomes (e.g., 'ADD' vs. 'ADD'), the calculation will be different. In practice, however, 'ADD' and 'ADD' would be considered the same permutation when drawn from the same set of letters, particularly in a scenario where the order of the letters does not confer a different outcome.