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

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...

• $10$ choose $3= \dfrac{10!}{7!\, 3!}=120$ – J. W. Tanner Jul 4 at 0:08
• $^nC_r$ is an alternate notation for the binomial symbol $\binom nr=\frac {n!}{r!(n-r)!}$. – lulu Jul 4 at 0:09

$${}^{10} C_3$$ means "$$10$$ choose $$3$$" and is more frequently denoted by the binomial coefficient $$\binom{10}{3}$$.
• Instead of "more frequently" say "in more advanced math". That ${}^nC_r$ notation is probably in common use in the US for ages 12 to 18 or so. Along with a corresponding ${}^nP_r$ notation. Called "combinations and permutations". – GEdgar Jul 4 at 0:12
• though in the Britannica example, $n$ is subscript rather than superscript, @GEdgar – J. W. Tanner Jul 4 at 0:20
$$^{10}C_3 = {10 \choose 3} = \cfrac{10!}{7! 3!} = \cfrac{ \Gamma(11)}{\Gamma(8) \Gamma(4)} = 120$$ You’ve guessed the right value.