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

Step 1: Determine the number of terms for each digit

We are given a sequence of ten terms. The terms can be $0$, $1$, or $2$. We need to find the number of such sequences that contain exactly five $1$ 's and exactly three $2$ 's. First, let's identify the total number of terms and the count for each specified digit.

Step 2: Calculate the number of 0's

Since the total number of terms is $10$, and we have five $1$ 's and three $2$ 's, the remaining terms must be $0$ 's. We calculate the number of $0$ 's by subtracting the count of $1$ 's and $2$ 's from the total number of terms.

Step 3: Apply the multinomial coefficient formula

This problem involves arranging a set of items where some items are identical. The formula for permutations with repetitions (also known as the multinomial coefficient) is used here. We have $N$ total terms, with $n_1$ terms of type 1, $n_2$ terms of type 2, and $n_3$ terms of type 3.

Step 4: Calculate the factorial values

Before performing the division, we need to calculate the factorial of each number involved in the formula. This step breaks down the calculation into manageable parts.

Step 5: Compute the final result

Substitute the calculated factorial values into the multinomial coefficient formula and perform the division to find the total number of unique sequences.

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