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Value of the expression { C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })

Question

Value of the expression
$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + .... + (C_{0} + C_{1} + .... + C_{n - 1})$ is

2^{n - 1}

n 2^{n - 2}

n 2^{n - 1}

2 n

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Value of the expression

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

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Text solutionVerified

Simplifying, we get

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

= n [^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + ...^{n} C_{n}] - [^{n} C_{1} + 2^{n} C_{2} + 3^{n} C_{3} + ... n^{n} C_{n}]

= n 2^{n} - \frac{d ( 1 + x ) ^{n}}{d x} ∣_{x = 1}

= n 2^{n} - n 2^{n - 1}

= n 2^{n - 1}

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Question Text	Value of the expression $C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + .... + (C_{0} + C_{1} + .... + C_{n - 1})$ is
Answer Type	Text solution:1
Upvotes	150