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Binomial Theorem for Any Index

If n C 8= n C...

Question

If $^{n} C_{8} =^{n} C_{2},$ find $^{n} C_{2} .$

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Solution

Here $^{n} C_{8} =^{n} C_{2} \Rightarrow^{n} C_{8} =^{n} C_{n - 2}$

$[∵^{n} C_{r} =^{n} C_{n - r}]$

$\Rightarrow 8 = n - 2 [∵^{n} C_{x} =^{n} C_{y} \Rightarrow x = y]$

$\Rightarrow n = 10$

$∴^{n} C_{2} =^{10} C_{2} = \frac{10!}{2! 8!} = 45.$

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^{n} C_{8} =^{n} C_{2}

^{n} C_{2}

^{n} C_{8} =^{n} C_{2}

, find the value of

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^{n} C_{8} =^{n} C_{2}

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a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .

b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .

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then the value of

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a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .

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