The total number of 10-digit sequences formed by only \( \{0, 1, 2\} \), where 1 should be used at least 5 times and 2 should be used exactly three times, is:
Show Hint
For combinatorial counting problems, break the problem into smaller parts and calculate the number of ways to arrange each part.
We are asked to find the total number of 10-digit sequences formed using the digits \( \{0, 1, 2\} \), where:
- \( 1 \) is used at least 5 times,
- \( 2 \) is used exactly 3 times.
Let’s break down the process:
1. Place the 3 occurrences of 2:
We need to place 3 occurrences of the digit 2 in the 10-digit sequence. The number of ways to choose 3 positions for 2 in a sequence of 10 digits is:
\[
\binom{10}{3}
\]
2. Place the 1's:
After placing the 2’s, 7 positions remain. We need to place at least 5 occurrences of the digit 1 in these 7 positions. The number of ways to place 5 ones in 7 available positions is:
\[
\binom{7}{5}
\]
3. Fill the remaining 2 positions with 0’s:
After placing the 2's and 1's, the remaining 2 positions will automatically be filled with the digit 0. Since there is only one way to place 0's in these positions, this part does not require further calculation.
Thus, the total number of sequences is:
\[
\binom{10}{3} \times \binom{7}{5} = \binom{10}{3} \times \binom{7}{2} = \binom{10}{5} \times \binom{5}{3}
\]
Therefore, the correct answer is \( \binom{10}{5} \times \binom{5}{3} \).
