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If $(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{15} x^{15}$ then value of the expression $S = C_{2} + 2 C_{3} + 3 C_{4} + \dots + 14 C_{15}$ is

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a

13214+1

b

13214

c

13215

d

none of these

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detailed solution

Correct option is A

We add −1C0+0C1 to both the sides of S and let S1=S−1C0+0C1=S−1.Note that S1=−1C0+0C1+1C2+2C3+…+13C14+14C15 (1)Using Cr=Cn-r, we rewrite the above expression as S1=14C0+13C1+12C2+…+0C14+(−1)C15 (2)Adding (1) and (2), we get 2S1=13C0+C1+C2+…+C15=13215⇒ S1=13214 ⇒ S=13214+1

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