Student: So you say we're going to look for familiar fractals in Pascal's Triangle. What is Pascal's Triangle??
Mentor: Well, let's look at a simple problem that Pascal was interested in (in the seventeenth century). It involves probability and chance, concepts which Pascal helped to develop. Here's the problem:
What are the chances that a family of X children will have Y girls?
Let's look at a few possible values of X and Y. Here's a table:
| Number of Children | Number of Girls | |
| X = 0 | Y = 0 | The chances are 1 out of 1 in this trival case, because if we know that there are no children, we know there are no girls. We don't have to check Y = 1 or anything else since one or more girls in a family of no children is nonsensical. |
| X = 1 | Y = 0 | Let's see; if we know there is 1 child, then it is either a girl or a boy. So, the chances are 1 out of 2 the child is a girl -- and the same for a boy. |
| X = 1 | Y = 1 | Same as the last: one child, one girl: 1 out of 2. We don't have to check Y = 2 or higher because that can't happen. |
| X = 2 | Y = 0 | Now it gets more interesting! With two children we can have no girls (so two boys), an older girl and a younger boy, a younger girl and an older boy, or two girls. All the possibilities can be summarized as: BB, BG, GB, GG. The chances that there are no girls (Y = 0) is 1 out of 4. |
| X = 2 | Y = 1 | We summarized the situation above. One girl happens 2 out of 4 times. |
| X = 2 | Y = 2 | Again, look at the cases above. Two girls happens 1 out of 4 times. As before, we're done with families of 2 children because Y = 3 would be impossible. |
Student: It seems that we need to look at all the possible ways kids can line up in a family to find the answer.
Mentor: Good! Can you show what happens for 3 children?
Student: Let's see; How can 3 children happen?
BBB GBB GGB GGG
BGB GBG
BBG BGG
Mentor: Good! Let's put these numbers in a table, where the directions for reading the table are: row number is number of children and column number is number of girls. We'll leave the impossible entries blank.
| 0 | 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|---|
| 0 | 1 | |||||
| 1 | 1 | 1 | ||||
| 2 | 1 | 2 | 1 | |||
| 3 | 1 | 3 | 3 | 1 | ||
| 4 | 1 | 4 | 6 | 4 | 1 | |
| 5 | 1 | 5 | 10 | 10 | 5 | 1 |
Student: Each number in this table is the number of ways that many girls can happen? So the second 10 in row 5 means that there are 10 different ways to have three girls in 5 children?
Mentor: Yes. Can you find all of these arrangements?
Student: Let's see;
GGGBB GGBGB GGBBG GBGBG
GBBGG BGGGB BFGBG BGGGB
Mentor: Very Good! Now, can you tell me -- without writing out all of the arrangements -- what row 6 would look like? Here's a big hint. Rewrite the table as a triangle, and look at how each number is
related to the two above it:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Student: Cool! The next row should be 1, 6, 15, 20, 15, 6, 1 -- you just add the two above.
Mentor: Exactly. This is Pascal's Triangle. We really should call it Zhu Shijie's Triangle, since Zhu, a Chinese mathematician from the fourteenth century, discovered it three hundred years before Pascal.
There are many interesting number patterns in this triangle. Try Coloring Multiples and Coloring Remainders .
Student: So Pascal's Triangle is used for calculating the chances of having a certain number of girls in a family?
Mentor: That is just one of many uses.
- In combinatorics and counting, we can use these numbers whenever we need to know the number of ways we can choose Y things from a group of X things. For example, if we need to choose 3 people to work on a problem together and we have 5 people to choose from, there are "5 choose 3" or 10 different ways to do this -- Using the fifth row, third entry in the triangle. This is the same idea as for the "number of girls in a family" problem.
- In algebra, we can use these numbers to figure out what a binomial raised to a power will be. Suppose we are taking (A + B) and raising it to the power 5. How does this work? We need to remember that the power 5 can be thought of as repeated multiplication, so we must find
(A + B)(A + B)(A + B)(A + B)(A + B)
This would be a mess to do, since we would have to use the distributive property over and over again. BUT!!! Pascal showed how his triangle gives the numbers needed to write this out after multiplication. The power is 5 so use row 5 to get:
See how the coefficients (that's the math term for the numbers in front of the letters in the expression) are just the numbers from row 5 of the triangle? So we write the power combinations in order (think of the "number of girls" problem) all A's (no B's), 4 A's (1 B), three A's, etc., up to no A's and then use the table for the coefficients.