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Combination

If nC12 = n...

Question

If $^{n} C_{12} =^{n} C_{6}$ , then $^{n} C_{2}$ is equal to:

A

72

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B

153

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C

306

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D

2556

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Solution

The correct option is D $153$
Given $^{n} C_{12} =^{n} C_{6}$
or $^{n} C_{n - 12} =^{n} C_{6}$
$\Rightarrow n - 12 = 6$
$\Rightarrow n = 18$
$∴^{n} C_{2} =^{18} C_{2} = \frac{18 \times 17}{2 \times 1}$
$= 153$

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Q. If ⁿC₁₂ = ⁿC₆, then ⁿC₂ = ____________.

^{n} C_{12} =^{n} C_{8}

then is equal to

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} + . . .

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

,
then the value of

{(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}

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} + . . .

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

,
then the value of

{(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}

n = 12 m (m \in N),^{n} C_{0} - \frac{^{n} C_{2}}{(2 + \sqrt{3})^{2}} + \frac{^{n} C_{4}}{(2 + \sqrt{3})^{4}} - \frac{^{n} C_{6}}{(2 + \sqrt{3})^{6}} + . . = (- 1)^{m} {(\frac{\sqrt{p}}{1 + \sqrt{Q}})}^{n}

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