Your on a road trip, your on your second day of driving. You have warn out your options of things to keep you busy, you have watched 3 movies, read two books, done all the puzzles in your puzzle book, colored all the pictures in the coloring book you brought, played ten games of "eye spy with my little eye", annoyed the person next to you to no end, and done a little bit of that homework you were supposed to get done before you left. You obviously have no other option but to look our the window and think about life's biggest questions: I wonder what my odds are of getting that chocolate cookie at lunch time instead of one of those other six cookies are that I packed, I wonder how many different ways I could have picked 2 pair of my shoes out of those 4 pairs I have.
Okay maybe you would not wonder these questions, maybe that is just me that would wonder but the point is there is a way to find out the answer to those questions, using the wonderful world of math. Now you are probably thing, "she is crazy", after all what person in there right mind would use say "the wonderful world of math". Well I'm not crazy... mostly not crazy.... okay maybe a little crazy but this does not change that fact that there is a cool way to find the answer to these questions without having to draw a picture that will probably just confusing anyway. You don't believe me? That's fine, I will prove it to you.
So lets start with a question, I like shoes so lets stick to shoes. I am going on a road trip to Hawaii and my suitcase will only fit three out of my seven pairs of shoes ( I know that no girl would have less then twenty pairs of shoes but just go with me) how many different ways could I pick my shoes? To answer this question we must start by defining some mathy words but I promise they are not hard word, if I can learn them you can. First "n choose k" this simply means the number of ways to choose k objects from n distinct objects. It is in essence a formula to answer the exact question that I just asked.
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n choose k is more commonly written as
\( \tbinom nk \)![]()
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and the formula is written like this
\(\tfrac{n!}{k!\,(n-k)!}\). I know that looks really scary but it's not, I promise. The "!" just means that you multiply whatever number it is by the number that are less then it in desending order, for example:
\(10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1\) . If you are like me and not good at doing this kind of math in your head, you then pull out a calculator and do it there, then the scary number is no longer scary. With this I think we are ready to do the problem from the top.
Question: I have 7 pairs of shoes and I can only pick 3 pair, how many ways can I pick my 3 shoes?
\( \tbinom 73 =\tfrac{7!}{3!\,(7-3)!}\)
\( \tbinom 73 = \tfrac{7!}{3!\,(4)!}\)
\( \tbinom 73 = \tfrac{7*6*5}{3*2*1} \)
\( \tbinom 73 = 35 \)
Answer: There are 35 ways to pick 3 pairs of shoes from 7 total pairs of shoes.
See, I told you I could do it and it was pretty fun actually. Simple to just put numbers into a formula and get an answer isn't it. Okay well that's all the mathing I am going to impose on you today. Thank you for reading.