The coefficient of x 53 in - r -0100100 C r x -3100- r 2 r isA- 100 C51B- 100 C52C- -100 C53D- 100 C54

The correct option is C ∑_r=0^100 ^100C_r(x 3)^100 r2^r=[(x 3)+2]^100=(x 1)^100=(1 x)^100=∑_r=0^100 ^100C_r( x)^r =∑_r=0^100 ^100C_r( 1)^r ^100C_r x^r r = 0 ∴ ...

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Byju's Answer

Standard XII

Mathematics

General Term of Binomial Expansion

The coefficie...

Question

The coefficient of $x^{53}$ in $\sum_{r = 0}^{100}^{100} C_{r} (x - 3)^{100 - r} 2^{r}$ is

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Solution

The correct option is C

$\sum_{r = 0}^{100}^{100} C_{r} (x - 3)^{100 - r} 2^{r} = [(x - 3) + 2]^{100} = (x - 1)^{100} = (1 - x)^{100} = \sum_{r = 0}^{100}^{100} C_{r} (- x)^{r}$

$= \sum_{r = 0}^{100}^{100} C_{r} (- 1)^{r}^{100} C_{r} x^{r}$ r = 0

$∴ C o e f f i c i e n t o f x^{53} = (- 1)^{53}^{100} C_{53} = -^{100} C_{53}$

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The coefficient of x53 in ∑100r=0 100Cr(x−3)100−r2r is

The correct option is C ∑100r=0 100Cr(x−3)100−r2r=[(x−3)+2]100=(x−1)100=(1−x)100=∑100r=0 100Cr(−x)r =∑100r=0 100Cr(−1)r 100Cr xr r = 0 ∴ Coefficient of x53=(−1)53 100C53=−100C53

The coefficient of $x^{53}$ in $\sum_{r = 0}^{100}^{100} C_{r} (x - 3)^{100 - r} 2^{r}$ is