First slide

Combinations

Question

The product of r consecutive positive integers is divisible by

Moderate

A

r!
B

(r – 1)!
C

(r + 1)!
D

None of these

Solution

Let r consecutive positive integers be
m, m + 1, m + 2, … (m + r – 1), where m ∈ N
$\begin{matrix} ∴ Product = m (m + 1) (m + 2) \dots (m + r - 1) \\ = \frac{(m - 1)! m (m + 1) (m + 2) \dots (m + r - 1)}{(m - 1)!} \\ = \frac{(m + r - 1)!}{(m - 1)!} = r! \cdot \frac{(m + r - 1)!}{r! (m - 1)!} \\ = r! \cdot^{m + r - 1} C_{r} \end{matrix}$
which is divisible by r! (∵ ^{m + r – 1}C_r is a natural number)

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