F 1- x n - r -0 n n C r x r and - r -0 n 1- n C r - a- then the value of -0 - i - n -0 - j - n i - n C i - j - n C j is equal toA- n2 a B- n2-0hig cD- na n -1

The correct option is D na(n+1)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) [ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Sum of Product of Binomial Coefficients

f 1+ x n =∑ r...

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{n a}{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

n a (n + 1)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

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

Suggest Corrections

Similar questions

Q. Assertion :If

f (x) = x^{n}

then

f (1) + \frac{f^{'} (1)}{1} + \frac{f^{''} (1)}{2!} + \frac{f^{'''} (1)}{3!} + . . . + \frac{^{n} f (1)}{n!} = 2^{n}

Reason: If

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

, then

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

Q. If

(1 + x)^{n} = C_{0} + C_{1} x + \dots . . + C_{n} x^{n},

then the value of

\sum \sum 0 \leq r < s \leq n (C_{r} + C_{s})

is equal to

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 evaluated value of

n \sum i = 0 n \sum j = 1

^{n} C_{j}

^{j} C_{i},

i \leq j

If $a_{n} = n \sum r = 0 \frac{1}{^{n} C_{r}}$ , then $n \sum r = 0 \frac{r}{^{n} C_{r}}$ equals:

Join BYJU'S Learning Program

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