Question

The number of four-letter words that can be formed using the letters of the word "BARRACK" is:

• A. 120
• B. 264
• C. 270
• D. 144

Hint:

When forming words with repeated letters, remember to divide by the factorial of the number of repetitions to avoid overcounting.

Answer 1

The Correct Option is C

We are asked to find the number of four-letter words that can be formed using the letters from the word "BARRACK". The word "BARRACK" consists of the letters: B, A, R, R, A, C, K. Thus, we have the following available letters: - B (1 time), A (2 times), R (2 times), C (1 time), K (1 time). We can form four-letter words by selecting four letters from the available ones. There are different cases depending on how many repeated letters we use. Case 1: No repeated letters. We can select 4 letters from the 5 distinct letters {B, A, R, C, K}. The number of ways to choose 4 letters from these 5 is given by: \[ \binom{5}{4} = 5. \] Then, we can arrange these 4 letters in any order, so the total number of words in this case is: \[ 5 \times 4! = 5 \times 24 = 120. \] Case 2: One letter repeated twice and two other distinct letters. We can select one letter to be repeated from {A, R}. There are 2 ways to choose which letter will be repeated. Then, we select 2 more letters from the remaining 4 distinct letters. The number of ways to do this is: \[ \binom{4}{2} = 6. \] Then, we arrange these 4 letters, considering that one letter is repeated twice. The number of distinct arrangements is: \[ \frac{4!}{2!} = 12. \] Thus, the total number of words in this case is: \[ 2 \times 6 \times 12 = 144. \] Case 3: Two letters repeated twice. We can select two letters to be repeated from {A, R}. There are 2 ways to choose the two repeated letters. Then, we arrange these 4 letters, considering that each letter is repeated twice. The number of distinct arrangements is: \[ \frac{4!}{2!2!} = 6. \] Thus, the total number of words in this case is: \[ 2 \times 6 = 12. \] Total number of four-letter words: Now, we add up the results from all the cases: \[ 120 + 144 + 12 = 270. \] Thus, the total number of four-letter words that can be formed using the letters of "BARRACK" is \( \boxed{270} \).