Counting Part 3: Buttons

Buttons (The Big Group)

Take 5 buttons: red, white, green, yellow, and purple. How many ways are there to choose 1 button from those 5? 5, of course.

How many ways are there to choose 2 buttons of different colors? The class agreed that order didn't matter - choosing red and white was the same thing as choosing white and red. The students found 10 ways to choose 2 buttons from 5.

We started in with the mathematical notation. The mathematical notation for "the number of ways to choose 2 things from 5" is 5C2. So we found

5C1=5

5C2=10

The most prevalent guess was that 5C3 would be 15, but it turned out that 5C3=10.

We found that 5C4=5, because choosing 4 buttons is the same as choosing 1 button to exclude. Here's what the board looked like after we figured all this out and started discussing what would happen if we started with 6 buttons:

The next day we continued finding 6C0, 6C2, ... , 6C6. Then we went back and started with 0C0. There's only one way to not take any buttons, which is to not take any. So

0C0=1

If we start with 1 button, we get

1C0=1, 1C1=1

With 2 buttons,

2C0=1, 2C1=2, 2C2=1

With 3 buttons,

3C0=1, 3C1=3, 3C2=3, 3C3=1

We continued down to row 6, and got the following pretty picture:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

This is a very famous creature called "Pascal's Triangle." To get each number for the next row, you add up the two numbers above it. For example, 5+10=15:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

On their own, each kid completed the triangle up to row 10 and looked for patterns. Some student observations:

• All the numbers in row 7 (except 1) are multiples of 7:

1 7 21 35 35 21 7 1

• All the numbers along the outside of the triangle are 1.
• The numbers that aren't quite along the outside go up by 1 each time:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

• To fill in the triangle, you only have to find half of a row because the second half mirrors the first:

1 6 15 20 15 6 1

• In rows that have a "middle number," the middle number is always even:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1