Question

If ${}^{13}{C}_r$ is denoted by $C_r$, then the value of $C_1+C_5+C_7+C_9+C_{11}$ is equal to

• A. $2^{12}-287$
• B. $2^{12}-165$
• C. $2^{12}-C_3$
• D. $2^{12}-C_2-C_{13}$

Answer 1

The correct option is A $2^{12}-287$

$C_1+C_3+C_5+C_7+C_9+C_{11}+C_{13}=C_0+C_2+C_4+C_6+C_8+C_{10}+C_{12}=2^{12}$

$\therefore$ $C_1+C_5+C_7+C_9+C_{11}=2^{12}-C_3-C_{13}$

$=2^{12}-\frac{13\times 12\times 11}{6}-1$

$=2^{12}-286-1=2^{12}-287$