I am stuck in a problem where for e.g., - I am calculating here $~^{10}C_4 + ^{10}C_6 + ^{10}C_8~$ where $~N~$ is same $(10$ in this example$)$ and $K$ is varying with difference of $2~($Odd or even indexed$)$, Here I have to individually calculate the Binomial Coefficient of each and then sum them up. Is there any formula that I can directly apply to achieve the same where I just need to put the range $([4-8]$ in the above example$)$ and get the result.
Key Points -
$ 1. ~~ N~~ $is Fixed
$ 2.~~ K~~$ is varying with difference of $2~ ($ It can be odd
indexed like $~^{10}C_3+^{10}C_5+^{10}C_7~$ or even indexed like $~^{10}C_2+ ^{10}C_4 +
^{10}C_6+\cdots~)$
$3.~~ K~~$ can start and end anywhere between $0$ to $N$.
Hope I am clear. Thanks in advance. Any help would be very nice.