In how many different ways can the letters of the word BANANA be arranged?

### Concept & Formula When a word has $n$ total letters but contains sets of identical letters, the number of distinct permutations is found by dividing the total factorial by the factorial of each repeating letter's frequency. $$P = \frac{n!}{p! \times q!}$$ ### Step-by-Step Solution - **Given:** The word 'BANANA'. - Count the total number of letters: $n = 6$. - Identify repeating letters: The letter 'A' appears $3$ times, and the letter 'N' appears $2$ times. - Set up the equation: Total valid arrangements = $\frac{6!}{3! \times 2!}$. - Expand and simplify: $\frac{6 \times 5 \times 4 \times 3!}{3! \times 2 \times 1}$. - Cancel out the $3!$ from numerator and denominator: $\frac{6 \times 5 \times 4}{2} = \frac{120}{2} = 60$. ### Exam Strategy & Shortcut Always cancel out the largest factorial in the denominator with the expanded numerator first. Doing $\frac{6 \times 5 \times 4}{2}$ mentally takes only a few seconds compared to computing $720 / 12$. ### Common Pitfall Only dividing by the factorial of one repeated letter (e.g., dividing only by $3!$ for the 'A's) and forgetting the 'N's entirely, which leads to the trap answer of $120$. ### Final Answer Therefore, the correct answer is **60**.