Final answer:
There are 84 different permutations that can be formed using 3 letters at a time from the word ADDITION, after considering the repetitions of the letters 'D' and 'I'.
Step-by-step explanation:
The word ADDITION consists of 8 letters, including some repetitions. To find the number of permutations that can be formed using 3 letters at a time, we need to consider the repeated letters. The number of times each letter appears in the word ADDITION is as follows:
- 'D' appears twice.
- 'I' appears twice.
- All other letters appear once.
When calculating permutations, you need to account for these repetitions. For a 3-letter permutation, the number of possible permutations is given by:
P(n, k) = n! / (n-k)! where n is the total number of letters, and k is the number of positions to fill. However, since there are repetitions, we must divide by the factorial of the number of repetitions for each repeated letter to avoid overcounting.
In our case, n = 8, and k = 3. The number of 3-letter permutations is calculated as:
P(8, 3) = 8! / (8-3)! = (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (5 x 4 x 3 x 2 x 1) = 8 x 7 x 6 = 336
Since the letters 'D' and 'I' are each repeated twice, we must divide 336 by 2! for each of these letters:
Corrected permutations = 336 / 2! / 2! = 336 / 2 / 2 = 84
Therefore, there are 84 different permutations that can be formed using 3 letters at a time from the word ADDITION.