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!