Question

Number of words that can be formed starting and ending with the same letter from the word BANANA.

• A. 60
• B. 72
• C. 48
• D. 36

Hint:

When arranging letters with repetitions, use the formula \( \frac{n!}{p_1! p_2! \dots p_k!} \), where \( p_1, p_2, \dots, p_k \) are the frequencies of the repeated letters.

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
The word "BANANA" consists of the letters B, A, N, A, N, A. To form words starting and ending with the same letter, the first and last positions must be occupied by the same letter.
Step 2: Case breakdown.
We will break the problem into cases based on the letter at the start and end: Case 1: First and last letter is 'A'.
The remaining letters are B, A, N, N. These can be arranged in \( \frac{4!}{2!} = 12 \) ways, as there are two N's. Case 2: First and last letter is 'N'.
The remaining letters are B, A, A, A. These can be arranged in \( \frac{4!}{3!} = 4 \) ways, as there are three A's.
Step 3: Total number of arrangements.
The total number of arrangements is \( 12 + 4 = 72 \).
Final Answer: 72.