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Maths-

General

Easy

Question

Let $C subscript 1 end subscript comma blank C subscript 2 end subscript comma blank C subscript 3 end subscript$ are the usual binomial coefficients. Let $S equals C subscript 1 end subscript plus 2 C subscript 2 end subscript plus 3 C subscript 3 end subscript plus... plus n C subscript n end subscript$ , then $S$ equals

${n 2}^{n}$
$2^{n - 1}$
${n 2}^{n - 1}$
$2^{n + 1}$

The correct answer is: $n 2 to the power of n minus 1 end exponent$

Let $S equals C subscript 1 end subscript plus 2 C subscript 2 end subscript plus 3 C subscript 3 end subscript plus horizontal ellipsis plus n C subscript n end subscript equals not stretchy sum from r equals 1 to n of blank r bullet blank to the power of n end exponent C subscript r end subscript$
$equals not stretchy sum from r equals 1 to n of blank r bullet fraction numerator n over denominator r end fraction blank to the power of n minus 1 end exponent C subscript r minus 1 end subscript blank open square brackets because blank to the power of n end exponent C subscript r end subscript equals fraction numerator n over denominator r end fraction blank to the power of n minus 1 end exponent C subscript r minus 1 end subscript close square brackets$
$equals n not stretchy sum from r equals 1 to n of blank blank to the power of n minus 1 end exponent C subscript r minus 1 end subscript$
$equals n open square brackets blank to the power of n minus 1 end exponent C subscript 0 end subscript plus blank to the power of n minus 1 end exponent C subscript 1 end subscript plus blank to the power of n minus 1 end exponent C subscript 2 end subscript plus horizontal ellipsis plus blank to the power of n minus 1 end exponent C subscript n minus 1 end subscript close square brackets$
$equals n 2 to the power of n minus 1 end exponent$

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