To find the number of sequences of ten terms, each being either 0, 1, or 2, containing exactly five 1s and exactly three 2s, follow these steps:

Determine the Remaining Terms: Since there are 5 ones and 3 twos, you will need 2 zeros to fill the sequence (because $5 + 3 + 2 = 10$).

Calculate the Number of Arrangements: You have a total of 10 positions to fill with these numbers (5 ones, 3 twos, 2 zeros). The formula for calculating permutations of a multiset is:

$ \frac{10!}{5! \times 3! \times 2!} $

Where:

$10!$ is the factorial of the total number of terms.

$5!$ is the factorial for the number of 1s.

$3!$ is the factorial for the number of 2s.

$2!$ is the factorial for the number of 0s.

Result: Simplifying the calculation gives:

$ \frac{10!}{5! \times 3! \times 2!} = 2520 $

Thus, there are 2520 possible sequences with the given conditions.