If n and r are two positive integers such that n- r- then nCr-1 - nCr -

The correct option is D ^n+1C_r+1 The value of ^nC_r+1+ ^nC_r =n!(n (r+1))!(r+1)!+n!(n r)!.r! =n!r!.(n (r+1))![1r+1+1n r] =n!r!.(n (r+ ...

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Standard XII

Mathematics

Sum of Coefficients of All Terms

If n and r ar...

Question

If n and r are two positive integers such that $n \geq r$ , then $^{n} C_{r + 1} +^{n} C_{r} =$

^{n} C_{n - r}

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^{n} C_{r}

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^{n - 1} C_{r}

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

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Solution

The correct option is D $^{n + 1} C_{r} + 1$
The value of $^{n} C_{r + 1} +^{n} C_{r}$
$= \frac{n!}{(n - (r + 1))! (r + 1)!} + \frac{n!}{(n - r)! . r!}$
$= \frac{n!}{r! . (n - (r + 1))!} [\frac{1}{r + 1} + \frac{1}{n - r}]$
$= \frac{n!}{r! . (n - (r + 1))!} [\frac{n + 1}{(r + 1) (n - r)}]$
$= \frac{(n + 1)!}{(n - r)! (r + 1)!}$
$=^{n + 1} C_{r + 1}$ .

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Similar questions

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. Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
(a)

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

(b) n ·_{^{n −}}_¹C_r_{− 1} = (n − r + 1) ⁿC_r_{− 1}
(c)

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

(iv) ⁿC_r + 2 · ⁿC_r_{− 1} + ⁿC_r_{− 2} = ^{n +}²C_r.

Q. If

(2 \leq r \leq n),

then

^{n} C_{r} + 2 \cdot^{n} C_{r + 1} +^{n} C_{r + 2}

is equal to

Q. If

n

and

r

are positive integers such that

r < n

, then

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

Q. Prove that

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

when

1 \leq r \leq n

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If n and r are two positive integers such that n≥r, then nCr+1+nCr=

The correct option is D n+1Cr+1The value of nCr+1+nCr=n!(n−(r+1))!(r+1)!+n!(n−r)!.r!=n!r!.(n−(r+1))![1r+1+1n−r]=n!r!.(n−(r+1))![n+1(r+1)(n−r)]=(n+1)!(n−r)!(r+1)!=n+1Cr+1.

If n and r are two positive integers such that $n \geq r$ , then $^{n} C_{r + 1} +^{n} C_{r} =$