The value of n C 0 + n + 1 C 1 + n + 2 C 2 + … .. + n + k C k is equal to

The given expression is the coefficient of x n in ( 1 + x ) n + ( 1 + x ) n + 1 + … . + ( 1 + x ) n + k ⇒ coefficient of x n in ( 1 + x ) n { 1 + x − 1 ( 1 + x ) k + 1 − 1 } = x ( 1 + x ) n ( 1 + x ) k + 1 − ( 1 + x ) n ⇒ Coefficient of x n + 1 in ( 1 + x ) n + k + 1 − ( 1 + x ) n i.e. n + k + 1 C n + 1 Or n + k + 1 C k

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Mathematics - Binomial Theorem Question with Solution | TestHub

MathematicsBinomial TheoremProperties of Binomial coefficientsMedium2 minQB

MathematicsMediummultiple choice

The value of $^{n} C_{0} +^{n + 1} C_{1} +^{n + 2} C_{2} + \dots .. +^{n + k} C_{k}$ is equal to

Options:(select one or more)

Answer:

A, B

Solution:

The given expression is the coefficient of $x^{n}$ in $(1 + x)^{n} + (1 + x)^{n + 1} + \dots . + (1 + x)^{n + k}$ $\Rightarrow coefficient of x^{n} in (1 + x)^{n} {\frac{( 1 + x ) ^{k + 1} - 1}{1 + x - 1}} = \frac{( 1 + x ) ^{n} ( 1 + x ) ^{k + 1} - ( 1 + x ) ^{n}}{x}$ ⇒ Coefficient of $x^{n + 1}$ in $(1 + x)^{n + k + 1} - (1 + x)^{n}$ i.e. $^{n + k + 1} C_{n + 1}$ $Or^{n + k + 1} C_{k}$

Stream:JEESubject:MathematicsTopic:Binomial TheoremSubtopic:Properties of Binomial coefficients

⏱ 2mℹ️ Source: QB

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