Friday, April 3, 2015

C is for . . .

I spent nine years focused on mathematics, first as an undergrad, then graduate student, then professor. After that? Diapers, playdates, and the quest for publication. It's been a wild ride, and during this A to Z Blog Challenge, I'm posting Mad-Cool-Math Nuggets as a nod to those crazy, number-filled days . . .


 Image courtesy: Sunchild57 Photography

C is for Counting.

There are lots of types of counting problems in discrete mathematics. Today I'll explain the difference between permutations and combinations. It's way more fun (and easy) than Calculus. Trust me.

Suppose you have 10 super favorite books. How many different ways could you arrange them on a bookshelf? Well, you could alphabetize by the author's last name, or alphabetize by the title, or arrange them in the order you read them. Lots of options, right?

Now let's get mathy. You have 10 choices to pick a book for, say, the leftmost position. Now there are 9 books left to choose from for the second slot. Then 8 books left to choose from for the third slot, and so on. So the answer is 10x9x8x7x6x5x4x3x2x1 = 10! = 3,628,800.

(Why do we multiply? Go here for the rule of product. What is that exclamation point for? It's short hand for factorial, the product of a number with all the natural numbers smaller than it down to 1. So 4! = 4x3x2x1)

Wow, there are over three million different ways to arrange 10 books! In this problem, order matters, and all those 3,628,800 options are called permutations.

Suppose order doesn't matter? Suppose you just want to pick 4 of those 10 books to take with you to the beach? This is a combination problem. The number of ways to pick 4 out of 10 objects turns out to be 210. I'll send you here if you really want to know the formula.

Fun fact: Combination locks are misnamed because order matters plenty with these things. If your "combination" is 2-3-4, dialing 3-4-2 won't work will it? They should be called permutation locks!

I still have nightmares that I'm in junior high and can't remember the permutation for my locker. How about you?


25 comments:

  1. Wow! I need to get my abacus out of retirement!

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  2. Tamara,

    Excellent and clear explanation. Thanks!

    Beth
    https://bethlapinsatozblog.wordpress.com/

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  3. Math was never my favorite subject. lol I sure didn't know that all of the options were called permutations.

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  4. Math was never my favorite subject. lol I sure didn't know that all of the options were called permutations.

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  5. Ah a numbers person, like my husband who even notices patterns in license plates as we are driving. I think it is genetic. I appreciate math and the large theories like the string theory, but cannot appreciate the relationships of numbers to one another. I am a terrible bridge player for that reason.

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  6. Yay for getting mathy!!! I have a math mind and the way to reach answers always feels intuitive to me, though I have two kids--one like me and the other who ISN'T so I get that for some people it is really not this way.

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  7. Yay for getting mathy!!! I have a math mind and the way to reach answers always feels intuitive to me, though I have two kids--one like me and the other who ISN'T so I get that for some people it is really not this way.

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  8. You know, math was never my favorite subject, but when you put it into a puzzle context like that, it is quite fascinating.

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  9. Math is great, sadly I am not that great at the more complex math. I can do the basics and with my calculator I can accomplish my work duties. My daughter that is away at college loves math. I do suffer from math envy since it is not my strongest ability. Thank you for teaching me something new :)!

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  10. Permutations and combinations are both simple and unique, I think. They can definitely be useful for laypeople wanting to make a quick estimation of probability and the like.

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  11. Math nerd! Hi-fi!

    Awesome blog post theme .. I'm gonna be here every day of April!

    Fellow #AtoZChallenge Blogger!
    Mithila @Fabulus1710
    http://www.fabulus1710.blogspot.in

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