What does the notation $^{10} C_3$ mean? Or: I am confused by a birthday card...

Question

I've just seen a gift card on the internet that is supposedly for a mathematician's 21st birthday. It says

$$ \text{Happy } ^{10} C_3 - 11\ln(e)-\frac{289}{3}+\left(\int_{\pi/6}^{\pi/3}\sec^2(x)dx\right)^2+7$$

I don't think I have ever seen the notation $^{10} C_3$ before. What does it mean?

Well,

$$ \left(\int_{\pi/6}^{\pi/3}\sec^2(x)dx\right)^2= \left(\tan(\pi/3)-\tan(\pi/6)\right)^2=\left(\sqrt{3}-\sqrt{1/3}\right)^2=4/3,$$

so I can simplify the expression that is wished to be happy to

$$ ^{10} C_3 - 11-\frac{289}{3}+4/3+7 = {}^{10} C_3 -99.$$

I guess the result is supposed to be 21, so (if I have not made any mistakes) I would expect that $${}^{10} C_3=120,$$ but I have absolutely no idea how to make sense of this...

Answer 1

${}^{10} C_3$ means "$10$ choose $3$" and is more frequently denoted by the binomial coefficient $\binom{10}{3}$.

Answer 2

$$^{10}C_3 = {10 \choose 3} = \cfrac{10!}{7! 3!} = \cfrac{ \Gamma(11)}{\Gamma(8) \Gamma(4)} = 120 $$ You’ve guessed the right value.