Find the coefficient of x n -2 inn C 0-n C 1 x - n C 2 x 2- - - n C n x n - n C 0- n C 1 x - n C 2 x 2- - - n C n x n A- n C0-n Cn-2-n C1-n Cn-3-n Cnn C2- n C0-n C2-n C1-n C3- -n Cn-2-n Cn- -rn-0n-n Cr-n Cn-2-rD- -r-0n Cr-n Cr-1

The correct options are A B C In the expansion of (^nC_0 + ^nC_1 x + ^nC_2 x^2 . . . . . ^nC_n x^n) × (^nC_0 + ^nC_1 x . . . . . ^nC_n x^n) We get the term ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

Find the coef...

Question

Find the coefficient of $x^{n - 2}$ in
$(^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n})$

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Solution

The correct options are
A

B

C

In the expansion of
$(^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} x . . . . .^{n} C_{n} x^{n})$
We get the terms with $x^{n - 2}$ when we multiply
$(^{n} C_{0} a n d^{n} C_{n - 2} x^{n - 2}), (^{n} C_{1} x a n d^{n} C_{n - 3} x^{n - 3}) . . . . . (^{n} C_{n - 2} x^{n - 2} a n d^{n} C_{0})$
From first sum and 2nd sum
⇒ The coefficient will be sum of each coefficient.
= $^{n} C_{0} \times^{n} C_{n - 2} +^{n} C_{1} \times^{n} C_{n - 3} . . . . .^{n} C_{n - 2} \times^{n} C_{0}$
This is what option (A) is. If we modify the terms using the property $^{n} C_{r} =^{n} C_{n} - r$
We get, $^{n} C_{0}^{n} C_{2} +^{n} C_{1} \times^{n} C_{3} + . . . . .^{n} C_{n - 2} \times^{n} C_{0}$
This is option (B)
Option (C) is the summation written with variable. When we expand (C) we get (B) and (A).

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Similar questions

Find the coefficient of $x^{n - 2}$ in
$(^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n})$

Q. If

(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{n} x^{n}, x ϵ N

then prove that

^{n} C_{0} -^{n} C_{1} +^{n} C_{2} - . . . . . . + (- 1)^{m - 1} \times^{n} C_{m - 1}

= (- 1)^{m - 1} \frac{(n - 1) (n - 2) . . (n - m + 1)}{(m - 1)!}

$^{n} C_{0} \times^{n + 1} C_{n} +^{n} C_{1}^{n} C_{n - 1} +^{n} C_{2} \times^{n - 1} C_{n - 2} + . . . . .^{n} C_{n}$ =

Coefficient of $x^{n}$ in $(^{n} C_{0} +_{1}^{C} x +^{n} C_{2} x^{2} . . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} \times . . . . .^{n} C_{n} x^{n})$ is

Q. If

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}

then which of the following is/are correct ?

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Find the coefficient of xn−2 in (nC0+nC1x+nC2x2.....nCnxn)×(nC0+nC1x+nC2x2.....nCnxn)

Find the coefficient of $x^{n - 2}$ in
$(^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n})$