Question

The number of arrangements of the letters of the word BANANA so that the arrangement starts and ends with the same letter, is

• A. 12
• B. 16
• C. 18
• D. 20
• E. 24

Hint:

Always list the letter counts first (B:1, A:3, N:2). This makes identifying possible start/end letters and the remaining permutations easier.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The word BANANA has 6 letters: B(1), A(3), N(2). We need to count cases where the first and last letters are identical.

Step 2: Detailed Explanation:
The identical letters available to place at the start and end are 'A' and 'N'.
Case 1: Starts and ends with 'A'
- Letters used: two 'A's.
- Remaining letters: $\{B, A, N, N\}$.
- Number of ways to arrange the remaining 4 letters:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \text{ ways} \]
(Dividing by $2!$ because there are two 'N's).
Case 2: Starts and ends with 'N'
- Letters used: two 'N's.
- Remaining letters: $\{B, A, A, A\}$.
- Number of ways to arrange the remaining 4 letters:
\[ \frac{4!}{3!} = \frac{24}{6} = 4 \text{ ways} \]
(Dividing by $3!$ because there are three 'A's).
Total ways: $12 + 4 = 16$.

Step 3: Final Answer:
The number of arrangements is 16.