Thursday, April 19, 2007

Scribe Post II

Hello everyone!!

We had an unusual start in today's class. When I first came inside the classroom, Mr. K was there, doing his daily morning routine which involves downloading the class' lesson for today. I left to go for a walk. When I came back, Mr. K wasn't there, and when the bell rung, to everyone's surprise, Miss Armstrong and her Grade 10 (Pre-Cal?) class came in. Miss Armstrong was saying that Mr. K is currently in her classroom probably using her computer. She then told us that we were going to do some "math aerobics" and whatnot. Everyone was hesitant at first, but after a while Miss Armstrong got us going and laughing. Richard tried to teach the sine and cosine dance to the Grade 10s and asked everyone to do it, but to no avail. Better luck next time, Richard, haha. ^_^ Mr. K then came in finally, and we started our lesson with the Grade 10s.

We then started off with this problem:
















When we did this problem for the first time, we did not include 0 as one of the possible even numbers for the 3rd problem in the slide. I was wondering at first as to why 0 wasn't included as an even number. That got Mr. K into thinking and we tried to find out whether 0 is an even number. After a couple of minutes, we found out that 0 is an even number and therefore was included as one of the possible even numbers, making us change our solutions and answers.
The solution should go like this:

1.1.5.4.5.1.5 = 500 phone numbers

We were then introduced to Permutations:

















This is pretty much straightforward. We were then introduced to the next problem:


















This problem was solved by one of the Grade 10s (his name is Daniel, thanks Aichelle) that were currently in our class. He figured out that since there were 9 players on the softball team, there could be 9.8.7.6.5.4.3.2.1 or 9! possible batting orders. The first member would have 9 choices when he would bat, leaving the second member 8 choices when to bat. The second member could then choose when he would bat, leaving the 3rd member 7 choices when to bat, so on and so forth. Daniel received a big compliment from Mr. K, who was impressed by the guy's thinking.

On to the next problem:
















This problem got most of us back into thinking, but it's actually quite simple. We could use the Fundamental Principle of Counting, or the Pick Formula. There are 120 rooms in a hotel, and guests have reservations for 66 different rooms. The first guest would have 120 rooms to choose from. After he chooses, the second guest has 119 rooms to choose from, 1 less than the first guest because, well, the first guest has chosen 1 room out of the 120 rooms available, leaving 119 rooms for the second guest to choose from. After the second guest chooses a room, the third guest chooses a room out of 118 rooms, then the fourth guest chooses a room out of 117 rooms, so on and so forth. There are at least 2.63x1012 ways the rooms can be assigned to guests.

Using the Pick Formula, nPr = n!/(n-r)!, we just plug in the numbers, as illustrated in the image.

n = 120 rooms
r = 6 guests

We would also arrive at the same answer. I actually prefer using the Fundamental Principle of Counting as it requires you to really think about it first and you would understand the concept more.

(note: Ms Armstrong came after we solved this problem and picked up the Grade 10s. The classroom felt empty after they left, haha.)

Then, the next problem Mr. K gave us got us into thinking. The word BOOK has four letters. How many "words" could we write using the letters from the word BOOK? Hmmm.. Grey-M said that there are 6 possible "words" that could be formed using the letters from the word BOOK. His reasoning is that since there are 2 letter O's in the word BOOK, some "words" would repeat. Mr. K then told us to actually see how many "words" we could come up using the letters from the word BOOK. Most, or all of us actually came up with 12 "words". Mr. K then explains us why in this slide.


















Why
12? Well, first of all there are four letters in the word BOOK, giving us 4! different "words" that can be made from that word. But, since there are 2 same letters, we divide 4! by 2! (Actually, I don't actually know yet as to why we divide by the 2! if there are 2 same letters , although I understand the concept. I just can't put into words yet. I'll explain further later when I could, haha.), therefore giving us 12 "words". Mr. K then gave us another example, this time he asked as to find how many "words" can be formed from the letters of MISSISSIPPI. Well:













That was pretty much our lesson for today. I'm just covering for Sandy who couldn't do the scribe responsibilities today, so she'll be the scribe for tomorrow. Homework for today is Exercise 29.

2 comments:

aichelle s. said...

good job John!

Lani said...

Hi John,

Thanks for your scribe!

May I ask a question? You said:
"Actually, I don't actually know yet as to why we divide by the 2! if there are 2 same letters , although I understand the concept. I just can't put into words yet. I'll explain further later when I could, haha.)" Are you at the point you can explain it yet? Do you think your understanding will be more rock solid if you can?

Best,
Lani