Using Pascal's Triangle
Suppose you are planning to have lunch. You have three items, a hamburger, an avocado, and some bananas that you can choose for lunch. You can choose to have one item for lunch, 2 items for lunch or all three. Or you can choose not to eat lunch. How many different possible lunches are there?
Consider the following table:
| Number of Items in Lunch | Possible Lunches | Number of Possible Lunches |
|---|---|---|
| 3 |  |
1 |
| 2 |  |
3 |
| 1 |  |
3 |
| 0 |  |
1 |
Altogether there are 8 lunches that are possible given 3 items to choose from.
Consider the same problem, but with four items to choose from. Suppose in addition to the hamburger, avocado and bananas, there are also some cherries:
| Number of Items in Lunch | Possible Lunches | Number of Posible Lunches |
|---|---|---|
| 4 |  |
1 |
| 3 |  |
4 |
| 2 |  |
6 |
| 1 |  |
4 |
| 0 | |
1 |
Altogether there are 16 possible lunches.
Note that if you only had one item available for lunch, you would have two possible lunches, and you can easily verify that if you had two items available for lunch, you would have 4 choices.
Consider the entries in Pascal's Triangle:
| 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 | ||||||||
| 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 |
Look at the entries in the 4th and 5th rows. They are the same numbers that we associated with the number of different lunches with three choices and four choices respectively.
That is, the entries in the 4th row are 1 - 3 - 3 - 1, and if we look at the 3 possible item lunches we get:
1 arrangement with 3 items
3 arrangements with 2 items
3 arrangements with 1 item
1 arrangement with 0 itemsSimilarly, the 5th row of Pascal's triangle corresponds to the numbers of arrangements when we worked with 4 items.
The numbers in Pascal's Triangle tell us how many arrangements of a number of objects there are with a fixed number of elements.
For example, the 7th row has entries:
| 1 | 6 | 15 | 20 | 15 | 6 | 1 |
This tells us that if we had a set with 6 different objects we could form:
1 set with 6 objects
6 sets with 5 objects
15 sets with 4 objects
20 sets with 3 objects
15 sets with 2 objects
6 sets with 1 object
1 set with 0 objectsTry a problem involving patterns from Pascal's Triangle.
email the author: Bruce Jacobs
Last modified: Thursday, February 6, 2003