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Series Involving Binomial Coefficients - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • Series Involving Binomial Coefficients is considered one of the most asked concept.

  • 123 Questions around this concept.

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In the expansion of (1+x)^5 the sum of the coefficient of the terms is

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In the expansion of (1+x)^n  the sum of coefficients of odd powers of x is

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If \\(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^nso, then the value of C_0+C_2+C_4+C_6+\ldots

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In the expansion of (1+x)^{50}, the sum of the coefficient of odd powers of x is

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In the expansion of (1+x)^5, the sum of the coefficient of the terms is.

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Concepts Covered - 1

Series Involving Binomial Coefficients

1. Sum of Binomial Coefficients 

\\C_{0}+\mathrm{C}_{1}+C_{2}+C_{3}+\ldots \ldots+C_{n}=2^{n}\\\text{or}\;\;\;\sum_{r=0}^{n} {^n} C_{r}=2^{n}
 

2. Sum of Binomial coefficients with alternate sign

\\C_{0}-\mathrm{C}_{1}+C_{2}-C_{3}+\ldots \ldots .+(-1)^{n} C_{n}=0\\\text{or}\;\;\;\sum_{r=0}^{n}(-1)^{r}\ ^{n} C_{r}=0

 

3. Sum of the Binomial coefficients of the odd terms & Sum of the Binomial coefficients of the even terms 

Adding the above two series we get,

\\2\cdot{(\mathrm{C}_{0}+\mathrm{C}_{2}+\mathrm{C}_{4}+\ldots \ldots)}=2^n\\\mathrm{C}_{0}+\mathrm{C}_{2}+\mathrm{C}_{4}+\ldots \ldots=2^{n-1}

Similarly, on subtracting we get,

\mathrm{C_{1}+C_{3}+C_5\ldots \ldots .}=2^{n-1}

Hence, the sum of the binomial coefficients of the odd terms = Sum of the binomial coefficients of the even terms 

I.e. \mathrm{C_{1}+C_{3}+C_5\ldots \ldots .=C_{0}+C_{2}+C_4+\ldots \ldots=2^{n-1}}

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Series Involving Binomial Coefficients

Study other Related Concepts

Series Involving Binomial Coefficients Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

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