If 2 n -1 C 0-2 n -1 C 1-2 n -1 C 2-02 n -1 C n -411 then which of the following is-are correct -A- n C 0- n C 1-2- n C 2-22- n C n -2 n -3-211B- n C 0- n C 1-2- n C 2-22- n C n -2 n -2-311- n C 0- n C 1-2- n C 2-22-1 n n C n -2 n -1-21111D- n C 0- n C 1-2- n C 2-22-1 n n C n -2 n -2-311

The correct options are A ^nC_0+^nC_12+^nC_22^2+⋯^nC_n2^n=(32)^11 C nbsp;^nC_0 ^nC_12+^nC_22^2 ⋯+( 1)^n^nC_n2^n=(12)^11^2n+1C_0+^2n+1C_1+^2n+1C_2+⋯+^2n+1C_n= ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

If 2 n +1 C 0...

Question

If $^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}$ then which of the following is/are correct ?

^{n} C_{0} + \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} + \dots \frac{^{n} C_{n}}{2^{n}} = {(\frac{3}{2})}^{11}

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^{n} C_{0} + \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} + \dots \frac{^{n} C_{n}}{2^{n}} = {(\frac{2}{3})}^{11}

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^{n} C_{0} - \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} - \dots + (- 1)^{n} \frac{^{n} C_{n}}{2^{n}} = {(\frac{1}{2})}^{11}

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^{n} C_{0} - \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} - \dots + (- 1)^{n} \frac{^{n} C_{n}}{2^{n}} = {(\frac{2}{3})}^{11}

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Solution

The correct options are
A $^{n} C_{0} + \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} + \dots \frac{^{n} C_{n}}{2^{n}} = {(\frac{3}{2})}^{11}$
C $^{n} C_{0} - \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} - \dots + (- 1)^{n} \frac{^{n} C_{n}}{2^{n}} = {(\frac{1}{2})}^{11}$
$^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}$
$\Rightarrow \frac{1}{2} (2^{2 n + 1}) = 2^{22}$
$\Rightarrow n = 11$

Now,
$^{n} C_{0} + \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} + \dots + \frac{^{n} C_{n}}{2^{n}} = n \sum r = 0 \frac{^{n} C_{r}}{2^{r}}$
$= n \sum r = 0^{n} C_{r} {(\frac{1}{2})}^{r} = {(1 + \frac{1}{2})}^{n} = {(\frac{3}{2})}^{n}$

$^{n} C_{0} - \frac{^{n} C_{1}}{2} + \frac{^{n} C_{2}}{2^{2}} + \dots + (- 1)^{n} \frac{^{n} C_{n}}{2^{n}} = n \sum r = 0 (- 1)^{r} \frac{^{n} C_{r}}{2^{r}}$
$= n \sum r = 0^{n} C_{r} {(\frac{- 1}{2})}^{r} = {(1 - \frac{1}{2})}^{n} = {(\frac{1}{2})}^{n}$

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Similar questions

{2.}^{n} C_{0} + 2^{2} . \frac{^{n} C_{1}}{2} + 2^{3} . \frac{^{n} C_{2}}{3} + \dots + 2^{n + 1} . \frac{^{n} C_{n}}{n + 1} =

\frac{^{n} C_{0}}{n} + \frac{^{n} C_{1}}{n + 1} + \frac{^{n} C_{2}}{n + 2} + \dots \dots + \frac{^{n} C_{n}}{2 n} =

$\frac{^{n} c_{0}}{n}$ + $\frac{^{n} c_{1}}{n + 1}$ + $\frac{^{n} c_{2}}{n + 2}$ +_ _ _ _ _ _ + $\frac{^{n} c_{n}}{2 n}$ =

$^{n} C_{0}$ + 2. $^{n} C_{1}$ + 3. $^{n} C_{2}$ +..............(n+1) $^{n} C_{n}$ =

$^{n} C_{0} \times^{n + 1} C_{n} +^{n} C_{1}^{n} C_{n - 1} +^{n} C_{2} \times^{n - 1} C_{n - 2} + . . . . .^{n} C_{n}$ =

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Sum of Coefficients of All Terms

Standard XII Mathematics

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If 2n+1C0+2n+1C1+2n+1C2+⋯+2n+1Cn=411 then which of the following is/are correct ?

If $^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}$ then which of the following is/are correct ?