Permutations & Combinations
5-digit numbers; product of digits = necklace count
MMTS_Full_Test_03
Grade 12
Question:
Number of 5-digit natural numbers such that product of their digits equals number of ways to create a necklace out of 6 beads from 10 distinct beads is
Step-by-Step Solution
Key Concept: Necklaces of 6 from 10 distinct beads: $\binom{10}{6}\cdot\frac{5!}{2}=\binom{10}{6}\cdot60=210\cdot60=12600$? Or $\binom{10}{6}\cdot(6-1)!/2=210\cdot60=12600$. Hmm, from solution: $x_1\cdot x_2\cdot x_3\cdot x_4\cdot x_5=9\cdot8\cdot7\cdot5\cdot5=12600$? And total 5-digit numbers = $5!/2\cdot...$=60.
60 numbers.
Correct Answer: 60