The value of -^15C_1+2^15C_2-3^15C_3+-15^15C_15+^14C_1+^14C_3+^14C_5++^14C_11 is

Question

The value of $-{}^{15}C_1+2\cdot{}^{15}C_2-3\cdot{}^{15}C_3+\cdots-15\cdot{}^{15}C_{15}+{}^{14}C_1+{}^{14}C_3+{}^{14}C_5+\cdots+{}^{14}C_{11}$ is

Accepted Answer

Let $S_1=\displaystyle\sum_{r=1}^{15}(-1)^r\,r\cdot{}^{15}C_r = 15\sum_{r=1}^{15}(-1)^r\,{}^{14}C_{r-1} = 15\left(-{}^{14}C_0+{}^{14}C_1-\cdots-{}^{14}C_{14}ight)=15	imes0=0$

Let $S_2={}^{14}C_1+{}^{14}C_3+\cdots+{}^{14}C_{11} = \left({}^{14}C_1+{}^{14}C_3+\cdots+{}^{14}C_{13}ight)-{}^{14}C_{13}=2^{13}-14$

$S_1+S_2=0+2^{13}-14=2^{13}-14$

Solutions · 1