Question

The product of $r$ consecutive positive integers is divisible by

• A. $r!$
• B. $(r-1)!$
• C. $(r+1)!$
• D. none of these

Answer 1

The correct option is A $r!$

Let the r consecutive positive integers be $m,m+1,m+2,...m+r-1$, where $m\in N$

Thus product $=m(m+1)(m+2)(m+3)...(m+r-1)$

or

$=\frac{(m-1!)m(m+1)(m+2)(m+3)...(m+r-1)}{(m-1!)}$

$=\frac{(m+r-1)!}{(m-1!)}=r!\frac{(m+r-1)!}{r!(m-1!)}r!{{}^{m+r-1}C}_r$

This is divisible by $r!$ as ${{}^{m+r-1}C}_r$ is a natural number.