# How many partitions $A_1,A_2,A_3$ are there of a set $S$, $|S| = 30$, and $|A_i| = 10$?

Let $$S$$ be a set with $$|S| = 30$$ and let $$\pi = \{A_i\}_{i=1}^3$$ be a partition of $$S$$ such that each set $$A_i$$ of $$\pi$$ has ten elements. How many such partitions $$\pi$$ are there?

This questions seems like a deceptively easy question, i.e., just a combination problem. Pick $${30\choose 10}$$ for $$A_1$$, $${30-10\choose 10}$$ for $$A_2$$, and $${30-20\choose 10}$$ for $$A_3$$ Then we have $${30\choose 10}{20\choose 10}{10\choose 10}$$. But we don't care in what order the 3 partitions are so finally answer is $$\frac{{30\choose 10}{20\choose 10}{10\choose 10}} {3!}.$$ Is this a sufficient answer?

• Yeah, it is ok.
– Aqua
Oct 13 at 20:39
• From the tag description for solution-verification: "This should not be the only tag for a question." Oct 13 at 20:41
• @Shaun, I'll fix it. Oct 13 at 20:41
• Thank you, @Owen. Oct 13 at 20:41

Your answer is correct. You can also write this in terms of a multinomial coefficient: $$\frac{\binom{30}{10,10,10}}{3!} = \frac{30!/10!^3}{3!}$$