Final answer:
The number of four-letter words that can be formed using the letters of the word BARRACK is 126. This total considers both the case with no repeated letters and the case where the letter 'R' is repeated, leading to 120 and 6 permutations respectively.
Step-by-step explanation:
To find the number of four-letter words that can be formed using the letters of the word BARRACK, we will consider various cases based on the repetition of letters. The word BARRACK has 7 letters where 'R' is repeated twice, and the other letters are unique (B, A, C, K).
Case 1: No repetition of letters
If we select any four different letters, the number of such permutations is 4! (4 factorial), which is 4×3×2×1 = 24 different combinations for each selection of letters.
Case 2: One letter repeated
When we choose the letter 'R' twice, we have two other positions to fill with the remaining letters (B, A, C, K), which can be done in 4P2 ways (permutations of 4 items taken 2 at a time). However, as 'R' is repeated, we must divide by 2! to correct for overcounting permutations of 'R'. Thus we have 4P2 / 2! = (4×3) / 2 = 6 ways for each such selection.
Total Number of Words
To get the total number of words possible, we add the number of permutations from each case. Since we have 5 unique letters to choose from (excluding the repeated 'Rs'), we have 5C4 ways to select which 4 letters to use in Case 1. Hence, the total for Case 1 is 5C4×24 permutations.
We only have one variation for selecting 'R' twice in Case 2, so the total is simply 6 permutations for this case.
Therefore, the total number of four-letter words that can be formed is (5C4×24) + 6 = (5×24) + 6 = 120 + 6 = 126 four-letter words.