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?

Correct Option (B)

The problem requires determining the maximum exponent 'm' of 3 in the prime factorization of 15!. This is equivalent to finding the total number of times the prime factor 3 appears in the product $1 \times 2 \times 3 \times ... \times 15$1×2×3×...×15.

To find the exponent of a prime 'p' in $n!$n!, the standard method involves summing the floor values of $n$n divided by successive powers of $p$p:

$E_p(n!) = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + ...$Ep​(n!)=⌊pn​⌋+⌊p2n​⌋+⌊p3n​⌋+...

In this specific case, $n = 15$n=15 and $p = 3$p=3. We calculate the terms as follows:

• First term: $\left\lfloor \frac{15}{3} \right\rfloor = 5$⌊315​⌋=5. This counts the multiples of 3 (3, 6, 9, 12, 15), each contributing at least one factor of 3.
• Second term: $\left\lfloor \frac{15}{3^2} \right\rfloor = \left\lfloor \frac{15}{9} \right\rfloor = 1$⌊3215​⌋=⌊915​⌋=1. This accounts for multiples of 9 (only 9 within the range 1 to 15), which contribute an additional factor of 3 (since 9 = 3 x 3, one factor was already counted in the first term).
• Third term: $\left\lfloor \frac{15}{3^3} \right\rfloor = \left\lfloor \frac{15}{27} \right\rfloor = 0$⌊3315​⌋=⌊2715​⌋=0. Higher powers of 3 will also yield 0, as $3^k > 15$3k>15 for $k \ge 3$k≥3.

Summing these values: $5 + 1 + 0 = 6$5+1+0=6.

Therefore, the maximum value of m is 6.

Incorrect Options:

• Option (A) 7: This value is incorrect. It may result from an arithmetic error or a misapplication of the method for counting prime factors in a factorial.
• Option (C) 5: This value is incorrect. It would be obtained if only the first term $\left\lfloor \frac{15}{3} \right\rfloor$⌊315​⌋ was considered, thereby omitting the additional factor of 3 contributed by multiples of 9 (e.g., 9 contributes two factors of 3, but only one would be accounted for in the first term).
• Option (D) 4: This value is incorrect and represents a significant underestimation of the number of prime factors of 3 in 15!.