If 1-x n - C 0- C 1 x - C 2 x 2- C n x n-then the value of -0 - r - s - n C r - C s 2 isA- n -1 -2 n C n -22 nB- n -1 -2 n C n -22 nC- n -1 -2 n C n -22 nD- n -1 -2 n C n -22 n

The correct option is D (n 1)·^2 n C_n+ 2^2 n nbsp;We have, 0 ≤ r lt;s ≤ n∑∑(C_r + C_s)^2 nbsp; nbsp; =(C_0 +C_1)^2+(C_0 + C_2)^2+… +(C_0 + C_n)^2 ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

If 1+x n = C ...

Question

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},$
then the value of $\sum \sum 0 \leq r < s \leq n {(C_{r} + C_{s})}_{r}^{2}$ is

(n - 1) \cdot^{2 n} C_{n} - 2^{2 n}

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(n + 1) \cdot^{2 n} C_{n} + 2^{2 n}

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(n + 1) \cdot^{2 n} C_{n} - 2^{2 n}

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(n - 1) \cdot^{2 n} C_{n} + 2^{2 n}

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Solution

The correct option is D $(n - 1) \cdot^{2 n} C_{n} + 2^{2 n}$
We have,
$\sum \sum 0 \leq r < s \leq n {(C_{r} + C_{s})}_{r}^{2}$

$= {(C_{0} + C_{1})}_{0}^{2} + {(C_{0} + C_{2})}_{0}^{2} + \dots + {(C_{0} + C_{n})}_{0}^{2} + {(C_{1} + C_{2})}_{1}^{2} + {(C_{1} + C_{3})}_{1}^{2} + \dots . + {(C_{1} + C_{n})}_{1}^{2} + ⋮ + ⋮ + \dots + ⋮$
$+ {(C_{n - 2} + C_{n - 1})}_{n - 2}^{2} + {(C_{n - 2} + C_{n})}_{n - 2}^{2} + {(C_{n - 1} + C_{n})}_{n - 1}^{2}$

$= n (C_{0}^{2} + C_{1}^{2} + \dots + C_{n}^{2}) + 2 \sum \sum 0 \leq r < s \leq n C_{r} C_{s}$
$= n \cdot^{2 n} C_{n} + [2^{2 n} -^{2 n} C_{n}] = (n - 1) \cdot^{2 n} C_{n} + 2^{2 n}$

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Similar questions

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. If

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

then

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

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For all positive integers n, show that

$^{2 n} C_{n} +^{2 n} C_{n - 1} = \frac{1}{2} (^{2 n + 2} C_{n + 1})$

Q. Assertion :If 'n' is even then

^{2 n} C_{1} +^{2 n} C_{3} +^{2 n} C_{5} + . . . . . +^{2 n} C_{n - 1} = 2^{2 n - 1}

Reason:

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Q. If

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If (1+x)n=C0+C1x+C2x2+…+Cnxn, then the value of ∑∑0≤r<s≤n(Cr+Cs)2 is

The correct option is D (n−1)⋅2nCn+22n We have, ∑∑0≤r<s≤n(Cr+Cs)2 =(C0+C1)2+(C0+C2)2+…+(C0+Cn)2+(C1+C2)2+(C1+C3)2+….+(C1+Cn)2+ ⋮ + ⋮ +⋯+ ⋮ +(Cn−2+Cn−1)2+(Cn−2+Cn)2+(Cn−1+Cn)2 =n(C20+C21+…+C2n)+2∑∑0≤r<s≤nCrCs =n⋅2nCn+[22n−2nCn]=(n−1)⋅2nCn+22n

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},$
then the value of $\sum \sum 0 \leq r < s \leq n {(C_{r} + C_{s})}_{r}^{2}$ is