Permutations and Combinations: Lesson
Lesson 1 of 1
  1. 1
    Enumeration
    An enumeration is defined as the number of ways to select from, or arrange, a set of n objects. Before we can begin studying this process, we have to explore some fundamental concepts. Let’s …
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  2. 2
    Enumerations With Repetition
    An ice cream parlor offers 27 different flavors of ice cream! Customers are allowed to fill their ice cream cone with three flavors of their choosing. How many different ice cream treats can be con…
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  3. 3
    Permutation
    In this lesson, we will explore the concept of a permutation. A permutation is an arrangement or selection in which the order matters. Consider this scenario: a college student has just purchase…
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  4. 4
    Permutations of a Subset
    In the previous exercise, you learned how to compute the number of permutations of n items. What if you were only interested in the permutation of r out of those n items (permutation of four …
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  5. 5
    Combination
    Until now, we have considered enumerations where the order of arrangement or selection matters. In the last exercise, one flight itinerary chosen could be New York, Virginia, Florida, Louisiana, Te…
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  6. 6
    Permutations With Identical Items
    In this exercise, we will explore how to enumerate elements of a multiset (a set that contains elements that are not necessarily distinct). Consider the problem of having to determine the number …
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  7. 7
    Selecting Items From a Multiset
    In this exercise, we will explore problems in which we have to choose r items from n different varieties. Consider this example: a taco restaurant offers three options of taco to choose from. Y…
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  8. 8
    Element Composition
    When enumerating a multiset, we are sometimes interested in the relative order of some members of the set. Take for example the problem of figuring out how many ways we can order the letters in the…
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  9. 9
    Breaking a Problem Down Into Subcases
    So far, you have learned many techniques to enumerate members of a set. Well done! In a real-world problem, however, arranging or selecting from a set of items may not be so straightforward. A sing…
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  10. 10
    Set Exclusion
    If we want to find out the number of ways NOT to do something, we could break it down into the subcases that are allowed and then combine those. Sometimes this may be inefficient as there might be …
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  11. 11
    Review
    Congratulations on getting this far! We’ve covered a lot of topics in the fundamentals of combinatorics and as of now, you should have a solid understanding of the topic! Let’s briefly recap the im…
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