Problem Solving With Combinations
In this section, we will look at scenarios in which you wish to determine how many alternative combinations of any size you may select from a specified number of elements, some of which might be identical. Any collection of unique objects can be subjected to the same reasoning.
The total number of combinations containing at least one item chosen from a group of \(n\) distinct items can be represented algebraically as:
Remember that combinations are subsets of the group of n objects. A null set is a set that has no elements. Therefore, the null set is one of \(2^n\) subsets of a set with \(n\) unique members.
Example
In how many ways can a team with at least one player be selected from a squad of eight players?
The squad can select anywhere from 1 to 8 players, so \( n = 8 \). Since the team must have at least one player, we use the formula that excludes the null set:
\(\text{Combinations} = 2^n - 1\)
\(= 2^8 - 1\)
\(= 256 - 1\)
\(= 255\)
Therefore, we can determine there are 255 ways to form a team of at least one player from an eight-member squad.
Example
A picnic planner can select three types of sandwiches, two types of drinks, and two types of desserts to include in a picnic basket. If there has to be at least one selection, how many distinct picnic baskets can be made?
In order to calculate the total number of combinations, we can use the formula for independent choices where each type of item (sandwiches, drinks, desserts) can either be included or not:
\(2^3\) ways for sandwiches, \(2^2\) for drinks, \(2^2\) for desserts.
Next, we can multiply these possibilities to get the total combinations when choosing freely from any food item:
\(\text{Combinations} = (2^3 \times 2^2 \times 2^2)\)
\(= 8 \times 4 \times 4\)
\(= 128\)
Since the basket must contain at least one food item, we can subtract the one scenario where no item is chosen:
\(\text{Combinations} = 128-1\)
\(= 127\)
Therefore, we can detemrmine there are 127 different ways to assemble a picnic basket with at least one item. This confirms the total number of unique picnic setups excluding the null set.
To determine the number of possible bouquets from seven roses, six carnations, and four chrysanthemums, ensuring at least one flower is included, follow this calculation:
First, we can calculate the total number of combinations for each flower type independently, considering each can either be included or not:
\(2^7\) ways for roses, \(2^6\) for carnations, and \(2^4\) for chrysanthemums.
Next, we can multiply these possibilities to get the total combinations when choosing freely from any flowers:
\(\text{Combinations} = 2^7 \times 2^6 \times 2^4\)
\(= 128 \times 64 \times 16\)
\(= 131,072\)
Since the bouquet must contain at least one flower, we can subtract the one scenario where no flower is chosen:
\(\text{Combinations} = 131,072 - 1\)
\(= 131,071\)
Therefore, the florist can create 131,071 different bouquets when at least one flower is used.