Okay, let's try to figure out how to solve this problem. So, the question is about the Greek army with two types of soldiers: upper class and lower class. There are 5 upper class soldiers and 10 lower class soldiers. The battle of Thermopylae requires a force of 4 upper class soldiers and 8 lower class soldiers. We need to find out how many different battalions can be sent. 

Hmm, so first, I think this is a combinatorics problem. The key here is to determine the number of ways to choose 4 upper class soldiers out of 5 and 8 lower class soldiers out of 10. Then, since these are independent choices, we can multiply the two results to get the total number of possible battalions. 

Let me recall the formula for combinations. The combination formula is C(n, k) = n! / (k!(n - k)!), where n is the total number of items, and k is the number of items to choose. So, for the upper class soldiers, it's C(5, 4), and for the lower class soldiers, it's C(10, 8). Then, multiply them together. 

Wait, let me make sure. The problem says "different battalions," which would be different combinations of soldiers. Since the two types of soldiers are separate, the total number of battalions is the product of the combinations for each type. 

Calculating C(5,4) first. Let's compute that. 5! / (4! * (5 - 4)!) = (5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * 1) = 5 / 1 = 5. So, there are 5 ways to choose 4 upper class soldiers from 5. 

Now, for the lower class soldiers, it's C(10,8). Let's compute that. 10! / (8! * (10 - 8)!) = (10 * 9 * 8! ) / (8! * 2!) = (10 * 9) / 2! = 90 / 2 = 45. So, there are 45 ways to choose 8 lower class soldiers from 10. 

Therefore, the total number of different battalions is 5 * 45 = 225. 

Wait, but let me double-check the calculations. For C(5,4), since choosing 4 out of 5 is the same as leaving out 1, which is straightforward. So 5 choose 4 is 5. For 10 choose 8, that's the same as 10 choose 2, because C(n, k) = C(n, n - k). So, 10 choose 2 is (10 * 9)/2 = 45. So that's correct. 

Multiplying them gives 5 * 45 = 225. That seems right. 

I think that's the answer. But let me just think again if there's any other factor I might have missed. The problem states "different battalions," which implies that the order of soldiers doesn't matter, right? So, combinations are appropriate here. If the order mattered, we would use permutations, but since it's a battalion, which is a group, the order isn't important. So, combinations are correct. 

Also, the problem doesn't mention any restrictions on the soldiers beyond the numbers required. So, all combinations of 4 upper class and 8 lower class soldiers are allowed. 

Therefore, the total number is 5 * 45 = 225. 

I think that's solid. No mistakes here.
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