If 1-xn-C0-C1 x- - -Cn xn- then the value of 0- r-s- n- Cr Cs is equal to

We have, ∑_r=0^n∑_s=0^n C_r C_s=(∑_r=0^n C_r^2)+2 0≤ r lt;s≤ n∑∑ C_r C_s ⇒ nbsp; nbsp;2^2 n=^2 n C_n+2 0≤ r lt;s≤ n∑∑ C_r C_s ⇒ nbsp;0≤ r lt;s≤ n ...

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Sum of Coefficients of All Terms

If 1+xn=C0+C1...

Question

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

\frac{1}{2} [2^{2 n} -^{2 n} C_{n}]

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\frac{1}{4} [2^{2 n} -^{2 n} C_{n}]

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\frac{1}{2} [2^{2 n} +^{2 n} C_{n}]

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\frac{1}{2} [2^{n} -^{2 n} C_{n}]

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Solution

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Similar questions

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

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

c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

{(1 + x)}^{n}

c_{0} c_{r} + c_{1} c_{r + 1} + c_{2} c_{r + 2} + . . . . + c_{n - r} c_{n} = \frac{| 2 n - - -}{| n - r - ---- - | n + r - ---- -}

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

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + \dots (C_{0} + C_{1} + C_{2} + \dots + C_{n - 1})

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

(\frac{C_{0} + C_{1}}{C_{0}}) (\frac{C_{1} + C_{2}}{C_{1}}) (\frac{C_{2} + C_{3}}{C_{2}}) . . . . . . . . . (\frac{C_{n - 1} + C_{n}}{C_{n - 1}})

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . . . C_{n} x^{n}

C_{0} . C_{r} + C_{1} . C_{r} + C_{2} . C_{r + 2} + . . . . . . . + C_{n - r} . C_{n}

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