Prove that - nCrnCr-1-n-r-1r

^nC_r^nC_r 1=n!r!(n r)!×(r 1)!n!(n r+1)! =1r× (r 1)!(n r)! #160; (r 1)!(n r+1)(n r)! =n r+1r . ...

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Byju's Answer

Standard XII

Mathematics

nCr Definitions and Properties

Prove that : ...

Question

Prove that : $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$

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Solution

$\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n!}{r! (n - r)!} \times \frac{(r - 1)!}{n!} (n - r + 1)!$
$= \frac{1}{r \times (r - 1)! (n - r)!}$ $(r - 1)! (n - r + 1) (n - r)!$
$= \frac{n - r + 1}{r}$ .

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Similar questions

Q. Prove that

\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}

where

1 \leq r \leq n

Q. Prove that

\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}

when

1 \leq r \leq n

Let r and n be positive integers such that $1 \leq r \leq n .$ Then prove the following :

(i) $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$
(ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$
(iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$
(iv) $^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}$

Q. Prove that

^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}

Q. Prove that

^{n} C_{r} + {2.}^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}

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Prove that : nCrnCr−1=n−r+1r

nCrnCr−1=n!r!(n−r)!×(r−1)!n!(n−r+1)!=1r×(r−1)!(n−r)! (r−1)!(n−r+1)(n−r)!=n−r+1r.

Prove that : $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$

$\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n!}{r! (n - r)!} \times \frac{(r - 1)!}{n!} (n - r + 1)!$
$= \frac{1}{r \times (r - 1)! (n - r)!}$ $(r - 1)! (n - r + 1) (n - r)!$
$= \frac{n - r + 1}{r}$ .