If 15× 14× 13× … × 3× 2× 1 = 3^m× n where m and n are positive integers, then what is the maximum value of m?

To find the maximum value of 'm', we need to determine the highest power of 3 that divides 15 factorial (15!). This is equivalent to counting the total number of times the prime factor 3 appears in the product 15 × 14 × 13 × … × 3 × 2 × 1. We can do this by counting the multiples of 3, then the multiples of 3^2 (9), and so on, within the range of 1 to 15. 1. Count numbers from 1 to 15 that are multiples of 3: These are 3, 6, 9, 12, 15. There are 5 such numbers. Each of these contributes at least one factor of 3. 2. Count numbers from 1 to 15 that are multiples of 3^2 (which is 9): Only 9 is a multiple of 9. This number contributes an *additional* factor of 3 (since 9 = 3 x 3, one '3' was already counted when we considered multiples of 3). 3. Count numbers from 1 to 15 that are multiples of 3^3 (which is 27): There are no multiples of 27 within this range (as 27 > 15). Summing the counts: Total number of factors of 3 = (Number of multiples of 3) + (Number of multiples of 9) Total factors of 3 = 5 + 1 = 6. Therefore, the maximum value of m is 6. The final answer is B