Let C1, C2, C3... are the usual binomial coefficients where Cr= nCr . Let S=C1+2C2+3C3+...+nCn , then S is equal to

Question

Let C1, C2, C3... are the usual binomial coefficients where Cr= nCr . Let S=C1+2C2+3C3+...+nCn , then S is equal to (A) n 2n (B) 2n - 1 (C) n 2n - 1 (D) 2n + 1

Tardigrade Admin · Accepted Answer

Let S=C1+2C2+3C3+ ldots +nCn= Sigma textr = 1 textnr⋅ nCr = Sigma textr = 1 textnr⋅ (n/r) n - 1Cr - 1 [∵ n Cr = (n/r) n - 1 Cr - 1] =n Sigma textr = 1 textn n - 1Cr - 1 =n[ n - 1 C0 + n - 1 C1 + n - 1 C2 + ldots + n - 1 Cn - 1] =n2n - 1

Q. Let $C_{1}, C_{2}, C_{3} ...$ are the usual binomial coefficients where $C_{r} =^{n} C_{r}$ . Let $S = C_{1} + 2 C_{2} + 3 C_{3} + ... + n C_{n}$ , then $S$ is equal to

$n 2^{n}$

$2^{n - 1}$

$n 2^{n - 1}$

$2^{n + 1}$

Q. Let C1​,C2​,C3​... are the usual binomial coefficients where Cr​=nCr​ . Let S=C1​+2C2​+3C3​+...+nCn​ , then S is equal to

n2n

2n−1

n2n−1

2n+1

Let S=C1​+2C2​+3C3​+…+nCn​=Σr=1n​r⋅nCr​ =Σr=1n​r⋅rn​n−1Cr−1​[∵nCr​=rn​n−1Cr−1​] =nΣr=1n​n−1Cr−1​ =n[n−1C0​+n−1C1​+n−1C2​+…+n−1Cn−1​] =n2n−1

Q. Let $C_{1}, C_{2}, C_{3} ...$ are the usual binomial coefficients where $C_{r} =^{n} C_{r}$ . Let $S = C_{1} + 2 C_{2} + 3 C_{3} + ... + n C_{n}$ , then $S$ is equal to

$n 2^{n}$

$2^{n - 1}$

$n 2^{n - 1}$

$2^{n + 1}$