Question

${}^{30}C_0{}^{30}C_{10}-{}^{30}C_1{}^{30}C_{11}+\dots \dots {}^{30}C_{20}{}^{30}C_{30}=?$

• A. ${}^{30}C_{11}$
• B. ${}^{60}C_{10}$
• C. ${}^{30}C_{10}$
• D. ${}^{65}C_{55}$

Answer 1

The correct option is C

${}^{30}C_{10}$

Explanation for the correct option:

Step 1.Write expansion of ${\left(1+x\right)}^{30}$ and ${\left(x-1\right)}^{30}$:

$\Rightarrow {(1+x)}^{30}={}^{30}C_0+{}^{30}C_{1}x+{}^{30}C_2{x}^2+.....+{}^{30}C_{20}{x}^{20}+...{}^{30}C_{30}{x}^{30}$ …....(1)

$\Rightarrow {(x-1)}^{30}={}^{30}C_0{x}^{30}-{}^{30}C_1{x}^{29}+....+{}^{30}C_{10}{x}^{20}-{}^{30}C_{11}{x}^{19}+{}^{30}C_{12}{x}^{18}+...{}^{30}C_{30}{x}^0$ ….....(2)

Step2. Multiply equation (1) and (2):

$\Rightarrow {(x^2-1)}^{30}=({}^{30}C_0+{}^{30}C_{1}x+{}^{30}C_2{x}^2+.....)\times ({}^{30}C_0{x}^{30}-{}^{30}C_1{x}^{29}+....)$

Step 3. Equate the coefficients of $x^{20}$ on both sides:

$\Rightarrow {}^{30}C_0{}^{30}C_{10}-{}^{30}C_1{}^{30}C_{11}+{}^{30}C_2{}^{30}C_{12}-...+{}^{30}C_{20}{}^{30}C_{30}={}^{30}C_r{x}^{2r}$

$={}^{\mathbf{30}}\mathit{C}_{\mathbf{10}}$ $\left[\because 2r=20\right]$

Hence, Option ‘C’ is Correct.