Question

Assertion :If 'n' is even then ${}^{2n}{C}_1{+}^{2n}{C}_3{+}^{2n}{C}_5+.....{+}^{2n}{C}_{n-1}=2^{2n-1}$ Reason: ${}^{2n}{C}_1{+}^{2n}{C}_3{+}^{2n}{C}_5+.....{+}^{2n}{C}_{n-1}=2^{2n-2}$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
• B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
• C. Assertion is true but Reason is false,
• D. Assertion is false but Reason is true.

Answer 1

The correct option is D Assertion is false but Reason is true.

$(1+x{)}^{2n}={}^{2n}{C}_0+{}^{2n}{C}_1{x}^1+{}^{2n}{C}_2{x}^2+{}^{2n}{C}_3{x}^3+...+{}^{2n}{C}_{2n}{x}^{2n}$
$(1-x{)}^{2n}={}^{2n}{C}_0-{}^{2n}{C}_1{x}^1+{}^{2n}{C}_2{x}^2-{}^{2n}{C}_3{x}^3+...-{}^{2n}{C}_{2n-1}{x}^{2n-1}+{}^{2n}{C}_{2n}{x}^{2n}$
Adding, both, we get
$(1+x{)}^{2n}-(1-x{)}^{2n}=2[{}^{2n}{C}_{1}x+{}^{2n}{C}_3{x}^3+{}^{2n}{C}_5{x}^5+...+{}^{2n}{C}_{2n-1}{x}^{2n-1}]$
Substituting, $x=1$, we get
$2^{2n-1}={}^{2n}{C}_{1}x+{}^{2n}{C}_3{x}^3+{}^{2n}{C}_5{x}^5+...+{}^{2n}{C}_{2n-1}{x}^{2n-1}$
$2^{2n-1}=2[{}^{2n}{C}_{1}x+{}^{2n}{C}_3{x}^3+{}^{2n}{C}_5{x}^5+...+{}^{2n}{C}_{2n-1}{x}^{n-1}]$
$2^{2n-2}={}^{2n}{C}_{1}x+{}^{2n}{C}_3{x}^3+{}^{2n}{C}_5{x}^5+...+{}^{2n}{C}_{2n-1}{x}^{n-1}$
Hence the assertion is false.