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Greatest Binomial Coefficients

If C0,C1,C2...

Question

If $C_{0}, C_{1}, C_{2}, . . . . . . . . . . C_{n}$ are the binomial coefficients in the expansion of $(1 + x)^{n}$ . n being even, then
$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . . + C_{n - 1})$ is equal to

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Solution

$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . + (C_{0} + C_{1} + C_{2} + . . . . + C_{n - 1})$
$= n C_{0} + (n - 1) C_{1} + (n - 2) C_{2} + . . . . + 1. C_{n - 1}$
$= n [C_{0} + C_{1} + C_{2} + . . . . + C_{n}] - [0. C_{0} + 1. C_{1} + 2. C_{2} + . . . . + n . C_{n}]$
$= n [2^{n}] - [n 2^{n - 1}]$
$= n [2^{n} - 2^{n - 1}]$
$= n 2^{n - 1}$

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C_{0}, C_{1}, C_{2} . . . . . . . . . . ., C_{n}

are the binomial coefficient in the expansion of

(1 + x)^{n}, n

being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2}, . . . . . . . . . . . C_{n}

are the Binomial coefficients in the expansion

(1 + x)^{n} .

‘n’ being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . . (C_{0} + C_{1} + C_{2} + . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2} . . . ., C_{n}

denote the binomial coefficients in the expansion of

{(1 + x)}^{n}

\frac{C 1}{C 0} + 2 \frac{C 2}{C 1} + + 3 \frac{C 3}{C 2} + . . . . . + n \frac{C n}{C n - 1}

C_{0}, C_{1}, C_{2}, \dots, C_{n}

are Binomial Coefficient in the expansion of

{(1 + x)}^{n}

C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \dots + \frac{C_{n}}{n + 1}

C_{0}, C_{1}

C_{n}

are binomial coefficients in the expansion of

(1 + x)^{n}

C_{0} + C_{1} . \frac{x}{2} + C_{2} . \frac{x^{2}}{3} + - - - - + C_{n} . \frac{x^{n}}{n + 1} =

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