The product of any r consecutive natural numb | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the $r$ consecutive natural numbers be $(n+1), (n+2), \dots, (n+r)$.Their product is $P = (n+1)(n+2) \dots (n+r)$.We can write this as:$P = \f

Vedclass · Accepted Answer

$r!$

Vedclass · Answer

(A) Let the $r$ consecutive natural numbers be $(n+1), (n+2), \dots, (n+r)$.Their product is $P = (n+1)(n+2) \dots (n+r)$.We can write this as:$P = \frac{(n+r)!}{n!}$.Multiplying and dividing by $r!$, we get:$P = \frac{(n+r)!}{n! r!} 	imes r! = \binom{n+r}{r} 	imes r!$.Since $\binom{n+r}{r}$ is the number of ways to choose $r$ objects from $n+r$ objects, it is always an integer.Therefore, the product $P$ is always divisible by $r!$.

The product of any $r$ consecutive natural numbers is always divisible by

Similar Questions

If ${}^n C_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression ${}^n C_{r+1} + {}^n C_{r-1} + 2{}^n C_r$ equals

$^nC_r + 2^nC_{r-1} + ^nC_{r-2} = $

In how many ways can $5$ red and $4$ white balls be drawn from a bag containing $10$ red and $8$ white balls?

In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of at least $3$ girls?

Vedclass Test Series

Exam Paper Generator

Online Exam Module