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:
Let's look at a few possible values of X and Y. Here's a table:
OK, so what did we learn from this table? 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 So, no girls happens 1 way, one girl happens 3 ways, two girls happens 3 ways, and three girls happens 1 way. 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.
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 GBGGB GBGBG GBBGG BGGGB BGBGG BGGBG 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.
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.
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