N-n-r-r-n-n-r-1-r-1-n-1-r-n-r-1-

LHS→n!(n r)!r!+n!(n r+1)!(r 1)! →n!(n r+1)(n r+1)(n r)!r!+n!r(n r+1)!(r 1)!r →n!(n r+1)(n r+1)!r!+n!r(n r+1)!r! →n!(n r+1+r)(n r+1)!r! ...

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Byju's Answer

Standard XII

Mathematics

Properties of Inverse Function

n!n-r!r!+n!n-...

Question

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

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Solution

$L H S \to \frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!}$

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

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

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

$\to \frac{(n + 1)!}{(n - r + 1)! r!} = R H S$

$L H S = R H S$

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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:

n! / r! x (n-r)! + n! / (r-1)! x (n-r+1) = (n+1)! / r! x (n-r+1)!

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

If $1 \leq r \leq n$ then $n^{n - 1} C_{r - 1}$ is equal to .........

Q. Observe the following Lists

List I	List II
$A)^{n} C_{r} +^{n} C_{r - 1} =$	$1)^{n + 1} P_{r}$
$B)^{\frac{^{n} P_{r}}{^{n} P_{r - 1}} =}$	$2) \frac{n - r + 1}{r}$
$C)^{n} P_{r} + r^{n} P_{r - 1} =$	$3) n - r + 1$
$D) \frac{^{n} C_{r}}{^{n} C_{r - 1}} =$	$4) n + r - 1$
	$5)^{(n + 1)} C_{r}$

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n!(n−r)!r!+n!(n−r+1)!(r−1)!=(n+1)!r!(n−r+1)!

LHS→n!(n−r)!r!+n!(n−r+1)!(r−1)!→n!(n−r+1)(n−r+1)(n−r)!r!+n!r(n−r+1)!(r−1)!r→n!(n−r+1)(n−r+1)!r!+n!r(n−r+1)!r!→n!(n−r+1+r)(n−r+1)!r!→(n+1)!(n−r+1)!r!=RHSLHS=RHS

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