There are $8$ different symbols. A code is formed by selecting $5$ symbols such that each symbol can be used only once (that is, no symbol is repeated in the code). Also, the order of symbols matters. How many different codes can be formed?

We need to find the number of ways to form a $5$ -symbol code from $8$ distinct symbols where repetition is not allowed and the order of symbols matters. This is a permutation problem.
The number of permutations of $n$ distinct objects taken $r$ at a time is given by the formula:

$${}^n{P}_r=\frac{n!}{(n-r)!}$$

Here, $n=8$ and $r=5$:

$${}^8{P}_5=\frac{8!}{(8-5)!}=\frac{8!}{3!}$$

$${}^8{P}_5=\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1}$$

$${}^8{P}_5=8\times 7\times 6\times 5\times 4$$

$${}^8{P}_5=56\times 120=6720$$

Therefore, $6720$ different codes can be formed.