What makes a number binomial?: Binomial Theorem

From the binomial theorem,

$(x+1)^{10} = \binom{10}{0}x^{10} + \binom{10}{1}x^{9} + \binom{10}{2}x^{8} + ... + \binom{10}{10}$

Thus the coeffiecient of $x^{8}$ is $\binom{10}{2}$

Thus $\binom{10}{2} = \large\frac{10!}{8!2!} = \boxed{45}$

It can easily be solved by using the Pascal's triangle.

Find row 10. That's 1,10,45,120,210,252,210,120,45,10,1 The 8th digit in the row is the coefficient for the x^8 factor, in this case 45