<Question>
The Greek army contained two types of soldiers: the upper class and the lower class soldiers. If there were a total of 5 upper class soldiers, and 10 lower class soldiers in a certain part of Athens, and the battle of Thermopylae demands a force of 4 upper class soldiers and 8 lower class soldiers, how many different battalions can be sent?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many different battalions can be sent from the Greek army based on the given numbers of upper and lower class soldiers. Let me start by understanding the problem step by step.
First, the total number of upper class soldiers available is 5, and the lower class soldiers are 10. The battle requires a force of 4 upper class soldiers and 8 lower class soldiers. The question is asking how many different battalions can be formed with these requirements.

Hmm, right. So, this seems like a combinatorics problem where I need to calculate combinations for each class and then multiply them together. Let me recall that combinations are used here because the order in which we select the soldiers doesn't matter. For example, choosing soldiers A, B, C, D is the same as D, C, B, A for the battalion.

So, for the upper class soldiers, we have 5 soldiers and need to choose 4.

<prompt_1.1> Calculate the number of ways to choose 4 upper class soldiers from 5 using combinations. </prompt_1.1>

Now, for the lower class soldiers, there are 10 in total, and we need to choose 8. Again, using combinations here.

<prompt_1.2> Calculate the number of ways to choose 8 lower class soldiers from 10 using combinations. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
The formula for combinations is C(n, k) = n! / (k! * (n - k)! ), where n is the total number, and k is the number to choose. So, plugging in the numbers for the upper class: C(5, 4) = 5! / (4! * (5 - 4)! ) = (5 * 4 * 3 * 2 * 1) / ( (4 * 3 * 2 * 1) * (1) ) = 120 / (24 * 1) = 120 / 24 = 5. Wait, that's 5? Let me check that again. If there are 5 upper class soldiers and we need to choose 4, then there are 5 different ways to leave one soldier out. So, yes, that makes sense. So, C(5,4) is indeed 5.
</execution_1.1>

<execution_1.2>
But wait, choosing 8 out of 10 is the same as choosing 2 to leave out, right? Because 10 - 8 = 2. Sometimes it's easier to calculate the combination for the smaller number. Let me check both ways to make sure.

Calculating directly: C(10,8) = 10! / (8! * (10-8)! ) = (10*9*8! )/(8! *2*1) = (10*9)/2 = 90/2 = 45.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2:
Alternatively, using the other approach: C(10,2) = 10! / (2! *8! ) = (10*9)/2 = 45.
So, same result.

So, the number of ways to choose the lower class soldiers is 45.

Now, to find the total number of different battalions, we need to multiply the number of ways to choose the upper class soldiers by the number of ways to choose the lower class soldiers.
That's because each combination of upper class soldiers can be paired with each combination of lower class soldiers. So, total battalions = 5 * 45 = 225.

Wait, so is the answer 225 different battalions?
Let me just verify that there's no overlap or dependency between the selections. Since the upper and lower class soldiers are distinct groups, choosing different sets from each class are independent. Therefore, multiplying the combinations is the correct approach.

Alternatively, if I think about it, for each of the 5 upper class soldiers, we can pair them with 45 different lower class groups.

<prompt_2.1> Verify the multiplication of combinations for upper and lower class soldiers and conclude the solution. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
But actually, wait, it's not exactly each individual soldier, but the combinations. However, since combinations are groups, multiplying them gives all possible pairs of groups. So yes, 5 groups from the upper class and 45 from the lower, each combination is unique. So 5*45=225.

I think that's correct. Let me check if there's any other factor I might have missed. For example, is there a restriction on the total number of soldiers in the battalion? The problem says 4 upper and 8 lower, so the total is 12 soldiers. There's no mention of any other constraints, like having to include certain soldiers or excluding others. So, the total is just the product of the two combinations.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the total number of different battalions that can be formed is $\boxed{225}$.
</Final_answer>
</think>