Which lock would you rather crack? Justify your reasoning mathematically.
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Counting
Have you ever tried to figure out how many ways something can happen? Like how many different burritos you can create at Chipotle? Or how many combinations of numbers that can occur in a Power Ball jackpot? These questions and more can be answered using methods of counting, including the Fundamental Counting Principle (sometimes called the Multiplication Counting Principle), permutations and combinations.
Have you ever tried to figure out how many ways something can happen? Like how many different burritos you can create at Chipotle? Or how many combinations of numbers that can occur in a Power Ball jackpot? These questions and more can be answered using methods of counting, including the Fundamental Counting Principle (sometimes called the Multiplication Counting Principle), permutations and combinations.
Fundamental Counting Principle: If event \(A\) can occur \(m\) ways and event \(B\) can occur \(n\) ways, then the number of ways events \(A\) and \(B\) can occur is \(m\ \cdot n\) ways.
Example 1: You are going to Chipotle to have a burrito. Chipotle offers burritos on white or whole wheat tortillas, \(2\) types of rice, \(2\) types of beans, and \(6\) meat choices. Before you add any additional toppings, and assuming you must make one choice from each category, how many options are available?
You can answer this questions by creating a tree diagram that illustrates all the possibilities:
If you total the branches, you will see there are \(48\) options.
Or you can use the Fundamental Counting Principle to find the answer. You multiply the number of ways to choose a tortilla (\(2\)) times the number of ways to choose rice (\(2\)) times the number of ways to choose beans (\(2\)) times the number of options for meat (\(6\)).
\(2\cdot2\cdot2\cdot6=48\). There are \(48\) options BEFORE you start choosing the rest of the fillers!
There are other types of problems that can be solved with the Fundamental Counting Principle that involve situations involving repetition of objects or no repetition of objects. For example, can you repeat numbers in your phone number or not? The number of possible outcomes will differ depending on whether repetition is or is not allowed.
Example 2: One configuration of license plates in Illinois is \(3\) letters followed by \(3\) numbers. How many license plates are possible if letters and numbers can be repeated? How many license plates are possible if letters and numbers cannot be repeated?
There are \(26\) letters in the English alphabets and \(10\) single digit numbers from \(0 – 9\) to choose from.
If repetition is allowed: \(26\ \cdot26\cdot26\cdot10\cdot10\cdot10=26^3\cdot10^3=17,576,000\)
There are \(17,576,000\) different license plates if repetition is allowed.
If repetition is not allowed: \(26\ \cdot25\cdot24\cdot10\cdot9\cdot8=11,232,000\)
There are \(11,232,000\) different license plates if repetition is not allowed. (Note that if repetition is not allowed, the number of choices is reduced.)
Other situations that arise involve compound events, which just means that more than one outcome is acceptable. Compound events can involve the word OR, which means we add the number of outcomes for each separate event. When compound events involve the word AND, it means we multiply the outcomes (like we have been doing with the fundamental counting principle).
Example 3: License plates in Illinois can have \(3\) letters followed by \(3\) numbers OR \(2\) letters followed by \(4\) numbers. How many license plates are possible if letters and numbers can be repeated?
We can also solve these types of problems using permutations.
Permutations
Another method of counting we will use to determine the number of ways events can occur involves combinations.
Combinations
Quick Check
Mrs. Moore has \(28\) students in her class. How many different ways can she seat her students in the first row that has six seats?
Quick Check Solution
Mrs. Moore has \(28\) students in her class. How many different ways can she seat her students in the first row that has six seats?
Quick Check Solution