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My Understanding of Counting Methods
When we first started this unit, I had a really hard time determining whether to use permutations or combinations. The formulas and processes made sense to me, and I knew how to use them, I more struggled with figuring out when to use them. Often, the wording of the questions sounded very similar to me, and I had trouble identifying the correct counting method to use. I went through this struggle when solving the problem pictured above as well, I thought it would be a permutation, because Julie and Marlowe have to be in the committee, therefore there is order, but that is false. Marlowe and Julie are the same as Julie and Marlowe, so the order does not matter (however if there were positions like President and Vice President involved this would change). However, as we continued to practice differentiating between when there is or isn't order I have become much more confident in my ability to identify which method to use and when. In fact, most of the time now I am able to tell which method to use right away, or I am able to catch my mistake before I get too far along in the problem.
I still get a little tripped up on some of the problems that involve both probability and counting methods, although I have found most of the time it is due to overthinking/overanalyzing the problem. As long as I am able to calm myself down and not overanalyze, I feel confident in my abilities to use counting methods.
8.3 Notes - Counting, Permutations, and Combinations
8.3 Notes - Counting, Permutations, and Combinations
This artifact is from the notes we took during the first half of Week Three, we had finished our first unit on the basics of probability and had now moved onto counting techniques and how to apply that to probability problems.
What are Counting Techniques (and factorials)?
Counting techniques are used when problems are too "big" to write out the entire sample space or create a probability tree. They allow us to determine the number of ways/the probability of something happening in a way that is quick and ensures we don't forget any possible outcomes.
Counting techniques utilize a math symbol called factorials. Factorials are defined as, "the product of decreasing whole numbers, beginning at n and ending at 1." Factorials are denoted as n!, where n represents any number, and the formula for a factorial is n! = n(n - 1)(n - 2)...(2)(1). An example of a factorial could be 5!, which would equal 5x4x3x2x1 or 120. The larger the n, the larger the product, that it why we use factorials, to keep these large equations simpler and more compact, I mean think of how 20! or 100! would look like written out, it would be huge.
Combinations vs. Permutations
There are two types of counting techniques that we used in class, combinations and permutations. Combinations are described as, "a collection of objects where the order does not matter," in other words we use combinations when counting a group of things that do not have set spots, like first, second, or third. The formula for using a combination is, nCr = (n/r) = n!/(n-r)!r!, where n is the number of objects and r is the number of objects used at a time.
Permutations are described as, "the arrangement of the objects in a specific order," in other words we use permutations when counting a group of things that have set spots, like president, vice president and secretary, or first place, second place and third place, and so on.
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