The value of (^21C_1-^10C_1)+(^21C_2-^10C_2)+(^21C_3-^10C_3)++(^21C_10-^10C_10) is:

Question

The value of $\left({}^{21}C_1-{}^{10}C_1\right)+\left({}^{21}C_2-{}^{10}C_2\right)+\left({}^{21}C_3-{}^{10}C_3\right)+\cdots+\left({}^{21}C_{10}-{}^{10}C_{10}\right)$ is:

Accepted Answer

$$\left({}^{21}C_1+\cdots+{}^{21}C_{10}\right)-\left({}^{10}C_1+\cdots+{}^{10}C_{10}\right)=\frac12\left[({}^{21}C_1+\cdots+{}^{21}C_{10})+({}^{21}C_{11}+\cdots+{}^{21}C_{20})\right]-(2^{10}-1)$$

$$=\frac12\left[2^{21}-2\right]-(2^{10}-1)=(2^{20}-1)-(2^{10}-1)=2^{20}-2^{10}$$

Solutions · 1