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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

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Result of Binomial Theorem -

Sum of product of binomial coefficients in the expansion is $\dpi{120} ^{2n}c_{n+r}$

hence $\dpi{120} c_{0}c_{r}+c_{1}c_{r+1}+----+c_{n-r}c_{n}= \frac{\left ( 2n \right )!}{\left ( n-r \right )!\left ( n+r \right )!}$

-

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

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

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

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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