If 1-xn-r-0nnCrxr and -r-0n1nCr-a- then the value of -0- i- n -0- j- ninCi-jnCj is equal to

Given, a=∑_r=0^n1^nC_r=∑_r=0^n1^nC_n r Let, y=∑_0≤ i≤ n ∑_0≤ j≤ n(i^nC_i+j^nC_j) = ∑_0≤ i≤ n ∑_0≤ j≤ n(n i^nC_n i+n j^nC_n j) [∵∑_r=0^nr^nC_r=∑_r=0^nn r^nC_n r ...

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Byju's Answer

Standard XI

Mathematics

Sum of Product of Binomial Coefficients

If 1+xn=∑r=0n...

Question

If $(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{r}$ and $n \sum r = 0 \frac{1}{^{n} C_{r}} = a,$ then the value of $\sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}})$ is equal to

n^{2} a

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\frac{n^{2}}{2 a}

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\frac{n a}{2}

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n a (n + 1)

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Solution

The correct option is D $n a (n + 1)$
Given, $a = n \sum r = 0 \frac{1}{^{n} C_{r}} = n \sum r = 0 \frac{1}{^{n} C_{n - r}}$
Let,
$y = \sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}})$
$= \sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{n - i}{^{n} C_{n - i}} + \frac{n - j}{^{n} C_{n - j}}) [∵ n \sum r = 0 \frac{r}{^{n} C_{r}} = n \sum r = 0 \frac{n - r}{^{n} C_{n - r}}]$
$= n \sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{1}{^{n} C_{n - i}} + \frac{1}{^{n} C_{n - j}}) - y = n \sum 0 \leq i \leq n (\frac{n + 1}{^{n} C_{n - i}} + a) - y$
$\Rightarrow 2 y = n {a \cdot (n + 1) + a \cdot (n + 1)}$
$∴ y = n a (n + 1)$

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Similar questions

Q. If

(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{r}

and

n \sum r = 0 \frac{1}{^{n} C_{r}} = a,

then the value of

\sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}})

is equal to

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f (x) = x^{n}

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Reason: If

{(1 + x)}^{n} = n \sum r = 0^{n} C_{r} x^{r}

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n \sum r = 0^{n} C_{r} = 2^{n}

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If (1+x)n=n∑r=0nCrxr and n∑r=01nCr=a, then the value of ∑0≤i≤n ∑0≤j≤n(inCi+jnCj) is equal to

If $(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{r}$ and $n \sum r = 0 \frac{1}{^{n} C_{r}} = a,$ then the value of $\sum 0 \leq i \leq n \sum 0 \leq j \leq n (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}})$ is equal to