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  • Help me understand! - The value of r for whichis maximum, is: - Binomial theorem and its simple applications - JEE Main

 

The value of r for which _{r}^{20}\textrm{C}\: \: _{0}^{20}\textrm{C}+_{r-1}^{20}\textrm{C}\: \: _{1}^{20}\textrm{C}+_{r-2}^{20}\textrm{C}\: \: _{2}^{20}\textrm{C}+................ is maximum, is:

  • Option 1)

     

    15

  • Option 2)

     

    11

  • Option 3)

     

    20

  • Option 4)

     

    10

Answers (1)

best_answer

 

Result of Binomial Theorem -

Sum of product of binomial coefficients in the expansion is ^{2n}c_{n+r}

hence c_{0}c_{r}+c_{1}c_{r+1}+----+c_{n-r}c_{n}= \frac{\left ( 2n \right )!}{\left ( n-r \right )!\left ( n+r \right )!}

 

-

 

 

\left ( 1+x \right )^{20}=^{20}\textrm{C}_0+^{20}\textrm{C}_1x+^{20}\textrm{C}_2x^{2}+\cdots +^{20}\textrm{C}_1x^{4}+\cdots

\left ( 1+x \right )^{20}=^{20}\textrm{C}_0+^{20}\textrm{C}_1x+^{20}\textrm{C}_2x^{2}+\cdots +^{20}\textrm{C}_r-1x^{r-1}+\cdots

\Rightarrow \left ( 1+x \right )^{40}=\sum \left ( ^{20}\textrm{C}_0\: \: ^{20}\textrm{C}_r+ ^{20}\textrm{C}_1^{20}\textrm{C}_r-1+\cdots \right )x^{r}+\cdots

Coefficient of x^{r}  is maximum in \left ( 1+x \right )^{40}  in r=20


Option 1)

 

15

Option 2)

 

11

Option 3)

 

20

Option 4)

 

10

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