A company is creating three new divisions and seven managers are eligible to be appointed head of a division. How many different ways could the three new heads be appointed?

$210$
Now, for the first division, we'll be able to select from any of the $7$ managers. For each of those $7$ potential options, we'll have $6$ remaining managers to choose for the second division head, and for each of those $6$, we'll have $5$ to choose from for the final division head.
In total, we'll have $7 \cdot 6 \cdot 5 = 210$ potential ways we could pick these heads of each of the three divisions from a pool of $7$ managers.