How many different signals can be made using 4 flags from a set of 7 different flags, if the order of the flags matters?

This is a permutation problem because the order of the flags matters for a signal (e.g., red-blue-green-yellow is different from blue-red-green-yellow). We are selecting a specific number of items from a larger set and arranging them.

We have 7 flags of different colors, so $n = 7$.
We are using exactly 4 flags for each signal, so $r = 4$.

The number of permutations of $n$ items taken $r$ at a time is given by the formula:

Substitute the values of $n$ and $r$:

Cancel out $3 \times 2 \times 1$ from the numerator and denominator:

There are $\boxed{840}$ different signals that can be made.