Matilda's Piggy Bank and Coin Exchange: An Exploratory Problem in Combinatorics

Welcome to an interesting exploration of combinatorics through the story of Matilda and her piggy bank. In this article, we will delve into the mathematical problem of determining how many different ways Matilda can pay for a game using the coins in her piggy bank. Through the application of combinatorial principles, we will break down the steps required for solving this problem and end with a clear, succinct answer.

Introduction to the Problem

Matilda has a piggy bank filled with 10 two-franc coins and 15 one-franc coins. She wishes to buy a game that costs 17 francs. The question arises: in how many different ways can Matilda pay for this game using the coins in her piggy bank?

Understanding the Problem Constraints

To approach this problem, we first need to understand the constraints provided:

Matilda has 10 two-franc coins. Matilda has 15 one-franc coins. The game costs 17 francs.

Given these constraints, we can establish that Matilda must use a combination of these coins to make up the total cost of 17 francs. The primary variable here is the number of two-franc coins, denoted as n.

Analyzing the Coins

Let's denote:

n as the number of two-franc coins used. The total amount of money Matilda uses from her piggy bank as 17 Francs, which is the cost of the game.

From the given coins, we need to find out how n can be chosen such that the total cost of 17 francs is met. The inequality that governs this is:

17 - 15  2n  17

Simplifying this, we have:

1 2n 2n 17

This gives us the range for n: n ( in [1, 8] )

or specifically, n can be any integer from 1 to 8 inclusive.

Counting the Number of Ways

Since n can take any integer value from 1 to 8, we can directly calculate the number of ways Matilda can use these two-franc coins to make up the remaining amount needed to reach 17 Francs. For each value of n, the number of one-franc coins required will be adjusted to meet the total amount.

Calculation for Each n

If n 1, then the remaining amount for one-franc coins is 17 - 2 15 francs. This can be paid all with 15 francs. If n 2, then the remaining amount is 17 - 4 13 francs. This can be paid in 13 francs with 13 one-franc coins. If n 3, the remaining amount is 17 - 6 11 francs. This can be paid in 11 francs with 11 one-franc coins. If n 4, the remaining amount is 17 - 8 9 francs. This can be paid in 9 francs with 9 one-franc coins. If n 5, the remaining amount is 17 - 10 7 francs. This can be paid in 7 francs with 7 one-franc coins. If n 6, the remaining amount is 17 - 12 5 francs. This can be paid in 5 francs with 5 one-franc coins. If n 7, the remaining amount is 17 - 14 3 francs. This can be paid in 3 francs with 3 one-franc coins. If n 8, the remaining amount is 17 - 16 1 franc. This can be paid in 1 franc with 1 one-franc coin.

In each case, there is exactly one way to pay the game with n two-franc coins and the corresponding one-franc coins. Therefore, the total number of ways Matilda can pay the game is the number of possible values for n, which is from 1 to 8. Hence, the number of ways is 8.

Conclusion and Final Answer

To summarize, the number of ways Matilda can pay for the game using her piggy bank coins is:

8

Thus, the final answer is 8 ways. This problem showcases the application of combinatorial principles and provides a clear, structured approach to solving such problems.