Question

prove n!/(n-r)! =n(n-1)(n-2)...(n-(r-1))

Answer 1

hope it helps . . . . . .

Answer 2

Answer:

In the question,

We have been given to prove that,

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

Therefore,

As we know from,

$n!=n(n-1)(n-2)......(2)(1)\\and,\\(n-r)!=(n-r)(n-(r-1))(n-(r-2)).........(1)$

Therefore, on putting the respective values of the terms in the given equation that we need to prove, we get,

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

Also,

$\frac{n!}{(n-r)!}=^{n}P_{r}$

As, Factorial is the product of consecutively decreasing numbers from the given number upto 1. We can say that the term's expansion can be done as shown.

Therefore,

The value is given as,

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

Hence, Proved.