The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to :

<p>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:</p> <p><p><strong>Determine the Remaining Terms</strong>: Since there are 5 ones and 3 twos, you will need 2 zeros to fill the sequence (because $5 + 3 + 2 = 10$).</p></p> <p><p><strong>Calculate the Number of Arrangements</strong>: 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:</p> <p>$ \frac{10!}{5! \times 3! \times 2!} $</p> <p>Where:</p></p> <p><p>$10!$ is the factorial of the total number of terms.</p></p> <p><p>$5!$ is the factorial for the number of 1s.</p></p> <p><p>$3!$ is the factorial for the number of 2s.</p></p> <p><p>$2!$ is the factorial for the number of 0s.</p></p> <p><p><strong>Result</strong>: Simplifying the calculation gives:</p> <p>$ \frac{10!}{5! \times 3! \times 2!} = 2520 $</p></p> <p>Thus, there are 2520 possible sequences with the given conditions.</p>