Pls help Q If O≤r, ≤n , then the coefficient of x' in the expansion of - Maths -

Question

Pls help

Q If O ≤ r ,   ≤ n , then the coefficient of x' in the
expansion of P = 1 + (1 + x) + (1 + x)2 + + (1 +
x)n is

(a) nCr â€‹ (b) n
+1Cr +1

(c) nCr +1 â€‹
(d) none of these

Aarushi Mishra · Accepted Answer

Let Px=1+1+x+1+x2+1+x3+
+1+xnThis is sum og G P
 with first term a=1 and common ratio r=1+xSum of n terms of G
P
=arn-1r-1Note here in Px we have n+1 terms starting from 1+x0 to 1+xnPx=11+xn+1-11+x-1=1+xn+1-1x=1+C1n+1 x + C2n+1 x2+C3n+1 x3+
+Crn+1 xr+Cr+1n+1 xr+1+
+Cn+1n+1 xn+1-1x=C1n+1 x + C2n+1 x2+C3n+1 x3+
+Crn+1 xr+Cr+1n+1 xr+1+
+Cn+1n+1 xn+1x=C1n+1  + C2n+1 x+C3n+1 x2+
+Crn+1 xr-1+Cr+1n+1 xr
+Cn+1n+1 xnThus general term of expression abovetr=Crn+1 xr-1We will get xr if we replace r by r+1 in general termtr+1=Cr+1n+1 xrCoefficient of xr is Cr+1n+1

Pls help Q. If O ≤ r , ≤ n , then the coefficient of x' in the expansion of P = 1 + (1 + x) + (1 + x)2 + ........ + (1 + x)n is (a) nCr ​ (b) n +1Cr +1 (c) nCr +1 ​ (d) none of these

Pls help

Q. If O $\leq r, \leq n$ , then the coefficient of x' in the expansion of P = 1 + (1 + x) + (1 + x)² + ........ + (1 + x)ⁿ is

(a) ⁿC_r (b) ^{n +1}C_{r +1}

(c) ⁿC_r ₊₁ (d) none of these