-r - 0n - 1 n cr n cr - n cr - i

The correct option is A n2 =^nC_r^nC_r + ^nC_r + 1 = ^nC_r^n+1C_r+1 = n!(n r)!r! . (n r)(r + 1)!(n + 1)! = r ...

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Byju's Answer

Standard XII

Mathematics

Greatest Binomial Coefficients

∑r = 0n - 1...

Question

$\sum_{r = 0}^{n - 1} \frac{^{n} c_{r}}{^{n} c_{r} +^{n} c_{r} + i}$

\frac{n}{2}

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\frac{n + 1}{2}

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\frac{n (n - 1)}{2 (n + 1}

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\frac{n (n + 1}{2}

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Solution

The correct option is A $\frac{n}{2}$
$= \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}} = \frac{^{n} C_{r}}{^{n + 1} C_{r + 1}}$
$= \frac{n!}{(n - r)! r!} . \frac{(n - r) (r + 1)!}{(n + 1)!} = \frac{r + 1}{n + 1}$
$\sum_{r = 0}^{n - 1} = \frac{1}{n + 1}$ [1 + 2 + 3 + ... + n]
$= \frac{n (n + 1)}{2 (n + 1)} = \frac{n}{2}$

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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. The value of

\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}

equals

Find the value of $\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}$ when n=100

Q. The value of

n - 1 \sum r = 0 \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}

is equals to ..............

Q. The value of

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

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∑n−1r=0ncrncr+ncr+i

The correct option is A n2=nCrnCr+nCr+1=nCrn+1Cr+1=n!(n−r)!r!.(n−r)(r+1)!(n+1)!=r+1n+1∑n−1r=0=1n+1[1 + 2 + 3 + ... + n]=n(n+1)2(n+1)=n2

$\sum_{r = 0}^{n - 1} \frac{^{n} c_{r}}{^{n} c_{r} +^{n} c_{r} + i}$