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What's the Difference? In English we use the word 'combination' loosely, without thinking if the order of things is important.

For example, saying: * 'My fruit salad is a combination of apples, grapes and bananas' We don't care what order the fruits are in, they could also be “bananas, grapes and apples” or “grapes, apples and bananas”, its the same fruit salad.

• 'The combination to the safe is 472'. Now we do care about the order; Entering the combination '724' won't work, nor will '247'. It has to be exactly 472 (in that order).

To avoid that type of confusion, in Mathematics we use more precise language:

• When the order doesn't matter, it is a combination.
• When the order does matter it is a permutation.

A 'combination lock' should more accurately be called a 'permutation lock'

BOWLS & CONES - On-Screen Math Activities (Ice Cream Problems)

• Bowls: There are six flavours of ice cream. If the ice cream is served in bowls that can hold up to six scoops, how many different ways can the ice cream be served?
• Cones: In a variation of the problem, the ice cream can be served in cones stacked up to

four scoops high. Given that the order of stacking matters, how many different cones could be served?

PROBLEM: The new pizza shop has been doing a lot of business. The owner thinks that it has been so hot this season that he would like to open up an ice cream shop next door.

He plans to start out with a small freezer and sell only six flavours of ice cream:

1. vanilla
2. chocolate
3. pistachio
4. boysenberry
5. cherry
6. butter pecan.

BOWLS: The cones that were ordered did not arrive in time for the grand opening so all the ice cream was served in bowls.

• How many choices for bowls of ice cream does the customer have?
• Find a way to convince each other that you have accounted for all possibilities.

CONES: The cones were delivered later in the week.

The owner soon discovered that people are fussy about the order in which the scoops are stacked. on the cone. One customer said “After all eating chocolate then vanilla is a different taste than eating vanilla then chocolate.”

The owner also quickly discovered that she couldn’t stack more than four scoops in a cone.

How many choices for ice cream cones does a customer have?

Find a way to convince each other that you have accounted for all possibilities.