For r = 0, 1, ... , 10, if Ar,Br and Cr denote respectively the coefficient of xr in the expansions of (1 + x)10, (1 + x)20 and (1 + x)30. Then, displaystyle∑r = 010 Ar(B10Br-C10Ar) is equal to

Question

For r = 0, 1, ... , 10, if Ar,Br and Cr denote respectively the coefficient of xr in the expansions of (1 + x)10, (1 + x)20 and (1 + x)30. Then, displaystyle∑r = 010 Ar(B10Br-C10Ar) is equal to (A) B10 -C10 (B) A10(B210-C10A10) (C) 0 (D) C10-B10

Tardigrade Admin · Accepted Answer

Ar = Coefficient of xr in (1 + x)10 = 10Cr Br= Coefficient of xr in (1+x)20 = 20Cr Cr= Coefficient of xr in (1+x)30 =30Cr ∴ displaystyle∑r=110 Ar (B10Br-C10Ar) = displaystyle∑r=110 Ar B10 Br- displaystyle∑r=110 Ar C10Ar = displaystyle ∑r=110 10Cr 20C10 20Cr - displaystyle ∑r=110 10Cr 30C10 10Cr = displaystyle ∑r=110 10C10-r 20C10 20Cr - displaystyle ∑r=110 10C10-r 30C10 10Cr =20C10 displaystyle ∑r=110 10 C10-r . 20Cr - 30C10 displaystyle ∑r=110 10 C10-r 10Cr =20C10 (30C10-1)- 30C10(20C10-1) = 30C10 - 20C10 = C10-B10

Q. For $r = 0, 1, ..., 10,$ if $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then, $r = 0 \sum 10 A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

$B_{10} - C_{10}$

$A_{10} (B_{10}^{2} - C_{10} A_{10})$

$0$

$C_{10} - B_{10}$

Q. For r=0,1,...,10, if Ar​,Br​ and Cr​ denote respectively the coefficient of xr in the expansions of (1+x)10,(1+x)20 and (1+x)30. Then, r=0∑10​Ar​(B10​Br​−C10​Ar​) is equal to

B10​−C10​

A10​(B102​−C10​A10​)

0

C10​−B10​

Q. For $r = 0, 1, ..., 10,$ if $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then, $r = 0 \sum 10 A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

$B_{10} - C_{10}$

$A_{10} (B_{10}^{2} - C_{10} A_{10})$

$0$

$C_{10} - B_{10}$