If C0, C1, C2, ldots ldots ldots ldots Cn denote the binomial coefficients in the expansion of (1+x)n, then the value of C0+(C0+C1)+(C0+C1+C2)+ ldots . +(C0+C1+ ldots .+Cn-1)

Question

If C0, C1, C2, ldots ldots ldots ldots Cn denote the binomial coefficients in the expansion of (1+x)n, then the value of C0+(C0+C1)+(C0+C1+C2)+ ldots . +(C0+C1+ ldots .+Cn-1) (A) n.2n-1 (B) n.2n (C) (n -1) .2n -1  (D) (n - 1) .2n

Tardigrade Admin · Accepted Answer

C0+(C0+C1)+(C0+C1+C2)+ ldots ldots ldots ldots +(C0+C1+ ldots ldots ldots ldots Cn-1) = n C0+(n-1) C1+(n-2) C2+ ldots ldots . Cn-1 =C1+2 C2+3 C3+4 C4 ldots ldots n Cn=n .2n-1

Q. If $C_{0}, C_{1}, C_{2}, \dots\dots\dots\dots C_{n}$ denote the binomial coefficients in the expansion of $(1 + x)^{n}$ , then the value of $C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + \dots .$ $+ (C_{0} + C_{1} + \dots . + C_{n - 1})$

$n . 2^{n - 1}$

$n . 2^{n}$

$(n - 1) . 2^{n - 1}$

$(n - 1) . 2^{n}$

$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + \dots\dots\dots\dots$
$+ (C_{0} + C_{1} + \dots\dots\dots\dots C_{n} - 1)$
$=^{n} C_{0} + (n - 1) C_{1} + (n - 2) C_{2} + \dots\dots . C_{n - 1}$
$= C_{1} + 2 C_{2} + 3 C_{3} + 4 C_{4} \dots\dots n C_{n} = n . 2^{n - 1}$

Q. If C0​,C1​,C2​,…………Cn​ denote the binomial coefficients in the expansion of (1+x)n, then the value of C0​+(C0​+C1​)+(C0​+C1​+C2​)+…. +(C0​+C1​+….+Cn−1​)

n.2n−1

n.2n

(n−1).2n−1

(n−1).2n