Summation notation for divided factorial.

Question

I have the following sum

$$5\cdot4\cdot3+5\cdot4\cdot2+5\cdot4\cdot1+5\cdot3\cdot2+5\cdot3\cdot1+$$$$5\cdot2\cdot1+4\cdot3\cdot2+4\cdot3\cdot1+4\cdot2\cdot1+3\cdot2\cdot1$$

It is basically $5!$ divided by two of the numbers in the factorial. So

$$\frac{5!}{1\cdot2}+\frac{5!}{1\cdot3}+\frac{5!}{1\cdot4}+...+\frac{5!}{3\cdot5}+\frac{5!}{4\cdot5}$$

Is there a way to write this as a single summation?

Answer 1

You can write it as a single sum as follows

$$\frac{5!}{1\cdot2}+\frac{5!}{1\cdot3}+\frac{5!}{1\cdot4}+...+\frac{5!}{3\cdot5}+\frac{5!}{4\cdot5}=5!\sum_{1\le i <j\le 5}\frac{1}{i\cdot j}.$$

Answer 2

Just as an alternative:

$$\sum_{S\subset \{1,2,3,4,5\}}_{|S|=3}\prod_{i\in S}i$$

This generalizes to other values.

It is also the coefficient of $x^3$ in the polynomial $(1+x)(1+2x)(1+3x)(1+4x)(1+5x)$, for what it is worth.