To solve this problem, let's break it down step by step.
### Step 1: Selecting and Arranging Children
We need to choose 2 out of 10 children and then arrange them.
1. Selecting 2 out of 10 children:
The number of ways to choose 2 children from 10 is given by the combination formula [tex]\( \binom{10}{2} \)[/tex]:
[tex]\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\][/tex]
However, in the context of permutations when selecting orders:
[tex]\[
P(10, 2) = \frac{10!}{(10-2)!} = 10 \times 9 = 90
\][/tex]
### Step 2: Selecting and Arranging Men
Next, we need to line up 4 out of 10 men, where the order matters.
2. Selecting 4 out of 10 men:
Similarly, we use the permutation formula [tex]\( P(10, 4) \)[/tex]:
[tex]\[
P(10, 4) = \frac{10!}{(10-4)!} = 10 \times 9 \times 8 \times 7 = 5040
\][/tex]
### Step 3: Selecting Women
For the women, we need to select 3 out of 10, but the order does not matter.
3. Choosing 3 out of 10 women:
The number of ways to choose 3 women from 10 is given by the combination formula [tex]\( \binom{10}{3} \)[/tex]:
[tex]\[
\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\][/tex]
### Step 4: Combining All Possible Ways
Finally, we multiply the number of ways to line up 2 out of 10 children, 4 out of 10 men, and choose 3 out of 10 women.
[tex]\[
\text{Total ways} = 90 \times 5040 \times 120
\][/tex]
Calculating the total:
[tex]\[
90 \times 5040 = 453600
\][/tex]
[tex]\[
453600 \times 120 = 54432000
\][/tex]
Thus, the total number of ways to accomplish this task is:
[tex]\[
54,432,000
\][/tex]
### Conclusion
Therefore, the correct answer is:
[tex]\[
\boxed{54,432,000}
\][/tex]
