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Combination

The least val...

Question

The least value of $n$ so that $^{n} C_{6} +^{n} C_{7} >^{n + 1} C_{6}$ is

A

13

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B

12

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C

11

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D

10

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Solution

The correct option is A 13
$^{n} C_{6} +^{n} C_{7} >^{n + 1} C_{6}$
$^{n + 1} C_{7} >^{n + 1} C_{6} \Rightarrow n + 1 > 7 + 6$
$=> n > 12$
$=> n = 13, 14, \dots$
$∴ n = 13$

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Q. The least positive integer

n

(n - 1) C_{5} + (n - 1) C_{6} = n C_{7}

n - 7

^{n - 1} C_{7} +^{n - 1} C_{6} >^{n} C_{6}

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_{7} =^{n} C_{4}

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