Prove that- n-n-r-nn-1n-2-n-r-1

We know that #160; n! = n× (n 1)× (n 2)× ...... 1 n! can be written as n (n 1)! #160; #160; #160; #160; #160; #160; #160; ...

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Standard XII

Mathematics

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Prove that: ...

Question

Prove that:
$\frac{n!}{(n - r)!} = n (n - 1) (n - 2) . . . (n - (r - 1))$

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Solution

we know that
$n! = n \times (n - 1) \times (n - 2) \times$ ...... 1
$n!$ can be written as $n (n - 1)!$
or $n (n - 1) (n - 2)$
generally $n! = n (n - 1) (n - 2) . . . . . (n - (r - 1)) (n - r)! \frac{n!}{(n - r)!}$
$= n (n - 1) (n - 2) . . . . . (n - (r - 1))$
hence proved.

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Similar questions

Q. Prove that:
(i)

\frac{n!}{(n - r)!}

= n (n − 1) (n − 2) ... (n − (r − 1))

(ii)

\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}

Prove that:
(i) $\frac{n!}{(n - r)!}$
= n(n-1)(n-2)...(n-(r-1))
(ii) $\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!}$
= $\frac{(n + 1)!}{r! (n - r + 1)!}$

Q. Prove that if

n

and

r

are positive integers

n^{r} - n {(n - 1)}^{r} + \frac{n (n - 1)}{2!} {(n - 2)}^{r} - \frac{n (n - 1) (n - 2)}{3!} {(n - 3)}^{r} + \dots

is equal to

0

r

be less than

n

, and to

n!

r = n

Q. Prove that

7^{n} {1 + \frac{n}{7} + \frac{n (n - 1)}{7.14} + \frac{n (n - 1) (n - 2)}{7.14.21} + . . . .}

= 4^{n} {1 + \frac{n}{2} + \frac{n (n - 1)}{2.4} + \frac{n (n - 1) (n - 2)}{2.4.6} + . . . .}

Q. If

(1 - x)^{- n} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . . . + a_{r} x^{r} + . . . . .

, then

a_{0} + a_{1} + a_{2} + . . . . . . + a_{r}

is equal to

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Prove that:n!(n−r)!=n(n−1)(n−2)...(n−(r−1))

we know that n!=n×(n−1)×(n−2)× ...... 1n! can be written as n(n−1)! or n(n−1)(n−2)generally n!=n(n−1)(n−2).....(n−(r−1))(n−r)!n!(n−r)! =n(n−1)(n−2).....(n−(r−1))hence proved.

Prove that:
$\frac{n!}{(n - r)!} = n (n - 1) (n - 2) . . . (n - (r - 1))$

we know that
$n! = n \times (n - 1) \times (n - 2) \times$ ...... 1
$n!$ can be written as $n (n - 1)!$
or $n (n - 1) (n - 2)$
generally $n! = n (n - 1) (n - 2) . . . . . (n - (r - 1)) (n - r)! \frac{n!}{(n - r)!}$
$= n (n - 1) (n - 2) . . . . . (n - (r - 1))$
hence proved.