How many positive numbers less than 10,000 have a product of digits equal to 210?

Step 1: Prime factorization of 210

First, find the prime factors of 210:
210 = 2 × 3 × 5 × 7

Step 2: Determine possible sets of digits

Numbers less than 10,000 can have 1, 2, 3, or 4 digits.
Since the product of digits is 210, none of the digits can be 0.
The digit 1 can be included without changing the product.

Possible valid sets of digits (all must be 1–9):

• Four-digit numbers: (7, 5, 3, 2)
• Three-digit numbers: (7, 6, 5) since 2 × 3 = 6

Step 3: Calculate permutations for each set

• Set (7, 5, 3, 2): 4 distinct digits → 4! = 24 permutations.
  All are valid 4-digit numbers.
• Set (7, 6, 5): 3 distinct digits → 3! = 6 permutations (3-digit numbers).
• Set (7, 6, 5, 1): For 4-digit numbers including 1 → 4! = 24 permutations.

Step 4: Sum the permutations

Total numbers = 24 + 6 + 24 = 54

Answer:

The correct answer is (D) 54.