Wednesday, April 30, 2008

Combinations and Permutations: The Difference

The Mastermind

Today's class started with Mr. K explaining the game shown above, "Mastermind". The concept is pretty simple, the mastermind, chooses any colours they want and arrange them in any order they want. Then the other player tries to guess the order each colour occurs. If you haven't already caught on, this is a permutation, because the order matters. If the other player guesses a colour correctly, the mastermind puts down a white peg, and if the other player guesses the right colour and places it in the right spot, then the mastermind puts down a black peg. But those pegs aren't placed in any particular order, so they're known as a combination, because order doesn't matter.

Did you get that? A permutation is an ordered arrangement of objects without repetition and a combination is an arrangement of objects where order doesn't matter. Using Mr. K's definitions of course. =)

After that, we watched a video that reviewed the past couple of days. It's called "Probability and Statistics" and is narrated by Ms. Jenkins. Watching this video felt like I was thrown back in grade seven, and sitting in my social studies class watching another educational video and absently taking notes. Hahahah. But it was helpful, so you should go watch it. Makes you want a tablet, hahaha.

ANYWAYS. The next few slides are just some problems that we worked on today. (I just realized that I'm not using the whole 'outline' format. Meh, too lazy to start over and.. meh, don't have time)

1. Use the 'bracelet' formula because a necklace is a circle. Circles have no beginning and no end and can be flipped over. Fill in your values.
(n-1)!/2
(12-1)!/2

2. Now solve!
11!/2
In your calculator, you would enter [11][MATH][<][4]
19958400

With this question we had slightly more trouble. I can't speak for everyone right now, but for me, what made this question confusing was the whole deal with alternating. I also didn't think of using 'slots' so I really made everything more difficult. But basically, mom and dad always has to be together. In this diagram, Paul (who came up to solve it) seated mom first and placed dad to her right. On her left side, one of the three boys have to sit there, and beside him, one of the three girls, and so on and so forth, filling in the slots as we went. Our original answer was 36. What we didn't realize was that dad could've have also sat on the left side of mom, which opens up a whole new set of numbers. So we changed the 1 in the second slot to a 2, therefore changing our answer to 72.

This question seemed easy enough, there are seven people and only three slots. So we thought, "well why not just fill in the blanks or use the 'pick' formula?"
7 x 6 x 5 = 210
7 P 3 = 210
We were all thinking, "that was too easy", and then Mr. K comes in and he's like, "are you sure?" Well, no he didn't really say that.. but he might as well have. THE POINT IS, it was wrong.

You see, when choosing who was going to be part of the committee, the order didn't matter. The 'pick' formula wouldn't apply. So he showed us another way of looking at it, basically, you're saying 'yes' to three people and 'no' to the rest (in this case, four). If you think about this as a word, you can just apply what was taught yesterday. There are two non-distinguishable objects, 'y' and 'n'.
7!/(3!4!) = 35

Lastly, we learned the 'choose' formula.

This is also in the slide if it's too small to see.

Aaannd.. that's the end of that. Thanks for waiting so long. I had to stop near the end to go to bible study at my church, and I almost lost everything earlier. SO YES, lots of problems today. But it's all good. xD

Oh, and the scribe shall be.. shall be.. (uhh, Mr. K, the scribe list is still not correct) Thi. Goodbye everyone!