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This problem involves determining how many different collections of 3 coins can be made from 5 types of coins: pennies, nickels, dimes, quarters, and half dollars. The coins are identical within each type, and the availability of each type is at least three coins.

### Explanation:
We need to find the number of ways to select 3 coins from 5 different types, where each type of coin is available in unlimited supply. This is a typical **"stars and bars"** problem in combinatorics, where we are distributing \( n \) identical items (coins) into \( k \) distinct groups (types of coins).

Here:
- The total number of coins to choose is \( n = 3 \).
- The number of different types of coins (groups) is \( k = 5 \).

The number of ways to distribute \( n \) indistinguishable items into \( k \) groups is given by the stars and bars formula:
\[
\binom{n+k-1}{k-1}
\]
Substituting \( n = 3 \) and \( k = 5 \):
\[
\binom{3 + 5 - 1}{5 - 1} = \binom{7}{4}
\]

Now, calculate \( \binom{7}{4} \):
\[
\binom{7}{4} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35
\]

### Final Answer:
The number of different collections of 3 coins that can be made is **35**.

Would you like a more detailed explanation of the stars and bars method or any other aspect of this solution?

Here are 5 related questions you might find interesting:
1. How would the result change if you had 4 coins to choose from instead of 3?
2. What if one type of coin was limited in supply (e.g., only 2 half dollars available)?
3. How does the stars and bars method apply to other scenarios, like distributing objects among people?
4. How many ways could you choose 5 coins from these types?
5. What if there were 6 types of coins instead of 5?

**Tip**: The stars and bars method is very useful for problems involving distributing indistinguishable items into distinct groups. It's a powerful tool in combinatorics!

How many different collections of 3 coins can be made if the coins can be pennies, nickels, dimes, quarters or half dollars, provided availability of each of the five types of coins is at least three and all coins of any given type are identical?

Learn how to find how many different collections of 3 coins can be made from 5 types (pennies, nickels, dimes, quarters, half dollars), with the stars and bars method. Explore key combinatorial concepts and the binomial coefficient formula. Suitable for Grades 10-12.

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