The coefficient of xr 0- r- n-1 in the expression x-2 n-1- x-2 n-2- x-1 - x-2 n-3- x-1 2- - x-1 n-1 is

The correct option is B ^nC_r ( 2^n r 1 ) Given expression is a G.P. #160; Therefore its sum is ( x+2 )^n 1{ ( x+1/x+2 )^n 1/ ( x+1/x+2 ) 1}= ( x+ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Binomial Coefficients

The coefficie...

Question

The coefficient of $x^{r} (0 \leq r \leq n - 1)$ in the expression ${(x + 2)}^{n - 1} + {(x + 2)}^{n - 2} \cdot (x + 1) + {(x + 2)}^{n - 3} \cdot {(x + 1)}^{2} + \cdot \cdot \cdot + {(x + 1)}^{n - 1}$ is

^{n} C_{r} (2^{r} - 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

^{n} C_{r} (2^{n - r} - 1)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

^{n} C_{r} (2^{r} + 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

^{n} C_{r} (2^{n - r} + 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $^{n} C_{r} (2^{n - r} - 1)$
Given expression is a G.P. Therefore its sum is
${(x + 2)}^{n - 1} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \frac{{(\frac{x + 1}{x + 2})}^{n} - 1}{(\frac{x + 1}{x + 2}) - 1} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = {(x + 2)}^{n} - {(x + 1)}^{n}$
Now coefficient of $x^{r} =^{n} C_{r} 2^{n - r} -^{n} C_{r} =^{n} C_{r} (2^{n - r} - 1)$

Suggest Corrections

Similar questions

Q. The coefficient of

x^{r} (0 \geq r \geq n - 1)

in the expansion of

(x + 2)^{n - 1} + (x + 2)^{n - 2} (x + 1) + (x + 2)^{n - 3} (x + 1)^{2} + . . . . . + (x + 1)^{n - 1}

^{n} C_{r} (2^{n - r} - 1)

Q. The coefficient of

x^{r} (0 \leq r \leq n - 1)

in the expansion of

E = {(x + 2)}^{n - 1} + {(x + 2)}^{n - 2} (x + 1) + {(x + 2)}^{n - 3} {(x + 1)}^{2} + . . . . + {(x + 1)}^{n - 1}

Q. The coefficient of

x^{r} [0 \leq r \leq n - 1]

in the expression of

(x + 2)^{n - 1} + (x + 2)^{n - 2} . (x + 1) + (x + 2)^{n - 3} . (x + 1)^{2} + . . . + (x + 1)^{n - 1}

Q. The coefficient of

x^{r} (0 \leq r \leq (n - 1))

in the expansion of

(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}

Q. if

a_{0}, a_{1}, a_{2}

....be the coefficients in the expansion of

(1 + x + x

^{2})^{n}

in ascending powers

f (x)

, then prove that :

(r + 1) a_{r + 1} = (n - r) a_{r} + (2 n - r + 1) a_{r - 1}, (0 < r < 2 n)

Join BYJU'S Learning Program

The coefficient of xr(0≤r≤n−1) in the expression (x+2)n−1+(x+2)n−2⋅(x+1)+(x+2)n−3⋅(x+1)2+⋅⋅⋅+(x+1)n−1 is

The correct option is B nCr(2n−r−1)Given expression is a G.P. Therefore its sum is(x+2)n−1⎧⎪ ⎪⎨⎪ ⎪⎩(x+1x+2)n−1(x+1x+2)−1⎫⎪ ⎪⎬⎪ ⎪⎭=(x+2)n−(x+1)nNow coefficient of xr=nCr2n−r−nCr=nCr(2n−r−1)

The coefficient of $x^{r} (0 \leq r \leq n - 1)$ in the expression ${(x + 2)}^{n - 1} + {(x + 2)}^{n - 2} \cdot (x + 1) + {(x + 2)}^{n - 3} \cdot {(x + 1)}^{2} + \cdot \cdot \cdot + {(x + 1)}^{n - 1}$ is