Question
Download Solution PDFWhat is \(\displaystyle\sum_{r=0}^n\)2r C(n, r) equal to ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Binomial expansion of (x + y)n is given by
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y2+.....+ nCn-1xyn-1 + nCnyn
Calculation:
Given, \(\displaystyle\sum_{r=0}^n\)2r C(n, r)
Expanding the expression,
⇒ nC020 + nC1 21 + nC222+.....+ nCn-12n-1 + nCn2n
⇒ nC01n 20 + nC1 1n-1 21 + nC21n-2 22+.....+ nCn-1 2n-1 + nCn2n
Comparing with binomial expansion x = 1 and y = 2
⇒ \(\displaystyle\sum_{r=0}^n\)2r C(n, r) = (1 + 2)n
⇒ \(\displaystyle\sum_{r=0}^n\)2r C(n, r) = 3n
∴ The correct option is (2).
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