If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . + C_{n} x^{n}$ , then $C_{o}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . + C_{n}^{2}$ is equal to

^{2 n} C_{n - 1}

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^{2 n} C_{n}

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^{2 n} C_{n + 1}

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none of these

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Solution

The correct option is B $^{2 n} C_{n}$
Given $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . + C_{n} x^{n}$ ...(i)
and ${(1 + \frac{1}{x})}^{n} = C_{0} + C_{1} \frac{1}{x} + C_{2} {(\frac{1}{x})}^{2} + . . . . + C_{n} {(\frac{1}{x})}^{n}$ ......(ii) (Replacing x by $\frac{1}{x}$ )
If we multiply (i) and (ii), we get
$(1 + x)^{n} {(1 + \frac{1}{x})}^{n} = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . + C_{n}^{2}$
$\frac{1}{x^{n}} (1 + x)^{2 n} = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . + C_{n}^{2}$
Clearly, RHS is the term independent of x and hence it is equal to the term independent of x in the product
$\frac{1}{x^{n}} (1 + x)^{2 n}$ or term containing $x^{n}$ in $(1 + x)^{2 n}$
Clearly the cofficient of $x^{n}$ in $(1 + x)^{2 n}$ occurs in $T_{n + 1}$ and is equal to $^{2 n} C_{n} = \frac{(2 n)!}{n! n!}$

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Similar questions

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}, t h e n C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . + C_{n}^{2}$ is equal to :

Q. If

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

then the value of

\sum \sum 0 \leq r < s \leq n {(C_{r} + C_{s})}_{r}^{2}

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. A.

^{2 n} C_{n} = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + \dots \dots + C_{n}^{2}

^{2 n} C_{n} =

term independent of

x

(1 + x)^{n} {(1 + \frac{1}{x})}^{n}

^{2 n} C_{n} = \frac{1.3.5.7 \dots \dots (2 n - 1)}{n!}

then

Find the value of $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} . . . . . C_{n}^{2}$ , where $C_{r} =^{n} C_{r}$

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If (1+x)n=C0+C1x+C2x2+....+Cnxn, then C2o+C21+C22+....+C2n is equal to

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . + C_{n} x^{n}$ , then $C_{o}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . + C_{n}^{2}$ is equal to