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Quantitative Aptitude

Minimum Value of A + B, If AB = Constant

If n C12=n C6...

Question

If ⁿC₁₂ = ⁿC₆, then ⁿC₂ = ____________.

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Solution

Given ⁿC₁₂ = ⁿC₆
Using ⁿC_r = ⁿC_n−r
⇒ n − 6 = 12
n = 18
$\begin{array}{rcl} \Rightarrow^{18} C_{2} & = & \frac{18 \times 17 \times 16!}{16! \times 2} \\ = & 9 \times 17 \end{array}$
i.e ¹⁸C₂ = 153

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^{n} C_{12} =^{n} C_{6}

^{n} C_{2}

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

then is equal to

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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