Review: Provide students with the following images, and explain the following:
Oreos have evolved largely over the last years, and now have many different types of cookies. The pictures show only some of the option available to consumers these days.
The pictures show:
- Chocolate or Vanilla flavoured cookies
- Strawberry, Chocolate, or Vanilla fillings
- A single layer, a double layer, or double layer with an additional cookie in between.
Since I cannot find any Oreo's with a different kind of cookie throughout, we shall assume all the cookies used have to be the same. Determine the total number of different Oreo's which could be created.
Provide students each with the following diagram
Ask them to determine the number of paths from various corners to other corners. **This might take some struggling...but let them struggle!**. Start with points close then further and further away. Leading them towards solving it by which ever method you prefer to teach.
Provide them with different paths and restrictions such as must start at XX go through XX and end up at XX.
Now introduce Combinations. In the recent Olympics, 8 men ran the 100m in the Final medal Race. First place recieved Gold, Second place recieved Silver, Third placed recieved Bronze. Determine the number of different ways the men could have placed.
Before the Finals there were heats were only the top 3 times in each heat would advance to the next level. If in the first heat, there were 8 men running, determine the number of different combination of men who could advance to the next round.
After the race, the men shake hands to congratulate each other. Determine the total number of handshakes for the 8 men.
Give students time to work together and lead them towards the idea, and then eventually explain, that when order DOES not make a difference, the number of combinations are
Permutations - order makes a difference - n!/(n-r)!
Combinations - order does not make a difference - n!/(n-r)!r!
How many different sums of money can you make with $5, $10, a penny, and a dime?