Sure, let's break down the problem step by step:
1. Understand the Problem:
- We have 10 identical coins.
- We need to determine how many ways we can arrange these coins so that we get exactly 5 heads and 5 tails.
2. Identify the Concept:
- This is a classic problem of combinations where we want to choose 5 positions out of 10 for either heads or tails. The remaining positions will automatically be filled by the other coin face.
3. Apply the Combinatorial Formula:
- The formula for combinations is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
- Here, [tex]\(n\)[/tex] is the total number of coins, which is 10, and [tex]\(k\)[/tex] is the number of positions we are choosing for heads (or tails), which is 5.
4. Calculate the Number of Combinations:
- Using the combination formula:
[tex]\[
\binom{10}{5} = \frac{10!}{5! \cdot 5!}
\][/tex]
This formula gives us the total number of ways to choose 5 positions out of 10 for heads.
5. Result:
- After performing the calculation (which involves factorials), we find that:
[tex]\[
\binom{10}{5} = 252
\][/tex]
So, the number of possible arrangements of the 10 coins that result in exactly 5 heads and 5 tails is 252.
