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?


To solve this problem, we need to calculate the number of possible combinations of soldiers that can be sent to the battle of Thermopylae. 

### Step 1: Calculate the combinations for upper class soldiers.
You have 5 upper class soldiers, and you need to choose 4 of them. The number of ways to do this is given by the combination formula:

\[
\text{Combinations of upper class soldiers} = \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1!} = 5
\]

### Step 2: Calculate the combinations for lower class soldiers.
You have 10 lower class soldiers, and you need to choose 8 of them. The number of ways to do this is given by:

\[
\text{Combinations of lower class soldiers} = \binom{10}{8} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
\]

(Note: \(\binom{10}{8}\) is equivalent to \(\binom{10}{2}\) because choosing 8 soldiers to include is the same as choosing 2 soldiers to exclude.)

### Step 3: Calculate the total number of different battalions.
Since the choice of upper class soldiers is independent of the choice of lower class soldiers, you multiply the two results:

\[
\text{Total number of battalions} = 5 \times 45 = 225
\]

So, **225 different battalions** can be sent to the battle.

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Do you want further details or have any questions? Here are 5 related questions:

1. How would the solution change if you needed to select 3 upper class and 7 lower class soldiers instead?
2. What if the total number of soldiers were different, say 6 upper class and 12 lower class?
3. How can you generalize this method for any number of upper and lower class soldiers?
4. What is the probability of selecting a particular battalion if all possible selections are equally likely?
5. How would you approach the problem if there were more than two classes of soldiers?

**Tip:** Remember that combinations are used when the order of selection does not matter, whereas permutations are used when the order does matter.

Learn how to calculate the number of different battalions that can be sent to the battle of Thermopylae using combinatorial mathematics. This problem involves selecting soldiers from two classes: upper class and lower class, with specific requirements. Suitable for high school students studying combinatorics and binomial coefficients.

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