The coefficient of x5 in the expansion of (1+x)21 +(1+x)22+ ldots+(1+x)30 is

Question

The coefficient of x5 in the expansion of (1+x)21 +(1+x)22+ ldots+(1+x)30 is (A)  31 C6- 21 C6 (B)  51 C 5 (C)  9 C5 (D)  30 C5+ 20 C5

Tardigrade Admin · Accepted Answer

We know that, coefficient of xr in the expansion of (1+x)n is given by n Cr. ∵ Coefficient of x5 in the expansion of (1+x)21+(1+x)22+ ldots+(1+x)2 n = 21 C5+ 22 C5+ ldots+ 30 C5 =( 21 C6+ 21 C5+ 22 C5+ ldots+ 30 C5)- 21 C6 -( 22 C6+ 22 Cb+ ldots+ 30 Cb)- 21 C6 [∵ n CY+ n Ct-1= n+1 CI] = . 23 C6+ 23 C5+ ldots+ 30 C6)- 21 C6 ...................................................... ...................................................... ...................................................... =( 30 C6+ 30 C5)- 21 C6 = 31 C6- 21 C6

Q. The coefficient of $x^{5}$ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$ is

$^{31} C_{6} -^{21} C_{6}$

$^{51} C_{5}$

$^{9} C_{5}$

$^{30} C_{5} +^{20} C_{5}$

Q. The coefficient of x5 in the expansion of (1+x)21+(1+x)22+…+(1+x)30 is

31C6​−21C6​

51C5​

9C5​

30C5​+20C5​

Q. The coefficient of $x^{5}$ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$ is

$^{31} C_{6} -^{21} C_{6}$

$^{51} C_{5}$

$^{9} C_{5}$

$^{30} C_{5} +^{20} C_{5}$