Introduction
In combinatorics, we often face problems that involve selecting subsets of items. A classic question is determining how many different sums can be formed using a set of items with distinct values. In this case, we have 4 coins, each with a different denomination.
The Logic
To determine the number of different sums, we must consider the choices available for each coin:
- For every individual coin, we have two choices: either we include it in our sum, or we exclude it.
- Since there are 4 coins, and for each coin we have 2 independent choices, the total number of ways to select a subset of these coins is $2^4 = 16$.
Accounting for the Empty Set
However, there is a catch. If we choose to "exclude" every single coin, we end up with a sum of 0. Depending on the context, a sum of 0 may or may not be considered a "sum of money." In standard mathematical combinatorics regarding non-empty subsets, we subtract this empty case.
- Total subsets = $2^4 = 16$
- Empty subset (no coins chosen) = 1
- Non-empty subsets = $16 - 1 = 15$
Because all 4 coins have different denominations, every non-empty subset of coins will produce a unique sum. Therefore, there are 15 possible different sums.
Step-by-Step Summary
- Identify the number of items ($n = 4$).
- Recognize that each item can be in state 'In' or 'Out'.
- Calculate the power set size: $2^n = 2^4 = 16$.
- Subtract the case where no coins are selected (the empty set): $16 - 1 = 15$.
- Conclude that 15 distinct sums can be formed.