Find the next number in the series:
$$6,24,120,720,\dots$$

Let's determine the operational pattern linking consecutive numbers in the given mathematical series:
* $6\times 4=24$
* $24\times 5=120$
* $120\times 6=720$

The series progresses by multiplying each term with an increasing sequential integer ($\times 4,\times 5,\times 6$). To find the subsequent value, we must multiply the final term by $7$:

$$\text{Next number}=720\times 7=5040$$