Question

${}^{10}{C}_1{+}^{10}{C}_3{+}^{10}{C}_5{+}^{10}{C}_7{+}^{10}{C}_9=$

• A. $2^9$
• B. $2^{10}$
• C. $2^{10}-1$
• D. None of these

Answer 1

Answer: (A)

$2^9$

For $(X+A{)}^n={C_0^{n}X}^n+\cdots C_r^n{X}^{n-r}{A}^r+\cdots C_n^n{A}^n$

Put $X=1;A=-1$

Hence, $0=C_0^n-C_1^n+C_2^n-C_3^n\cdots$ and so on

$\Rightarrow C_0^n+C_2^n{+C}_4^n+\cdots =C_1^n+C_3^n+C_5^n+\cdots$ ...(1)

Now put $A=1$ and we see that

$2^n=C_0^n+C_1^n+C_2^n{+C_3^n+C}_4^n+C_5^n+\cdots$ ...(2)

Hence, the sum of

$C_1^n+C_3^n+C_5^n+\cdots =2^{n-1}$ (From (1) and (2))

In the problem, $n=10$

Hence, answer is $2^9$