- Pascal's Triangle
- 1 Basic Description
- 2 A More Mathematical Explanation
- 2.1 Properties
- 2.2 More patterns within Pascal's triangle
- 2.3 Applications of Pascal's Triangle
- 2.4 In Combinatorics
- 3 Teaching Materials
Pascal's triangle is a triangular arrangement of specific numbers which have interesting patterns. We start out with 1 and 1, 1, for the first two rows. To construct each entry in the next row, we add the two numbers immediately above it to the right and to the left. If there's only one number diagonally above the entry to the left and to the right, then we enter just that number. We can continue doing this endlessly. This pattern is just one of the many patterns within the triangle. The hockey stick pattern below is another example of one of the most interesting and fun patterns within the triangle.
Hockey Stick Pattern
If we select entries diagonally starting from 1 on the edge down to any point in the triangle, the sum of the numbers selected is equal to the number below the selection that is not on the diagonal, forming a pattern that looks like a hockey stick. To understand this better, please look at the imgage on the left.
A More Mathematical Explanation
Each entry in a Pascal's triangle is identified by a row and a place. The rows are labeled starting f [...]
Each entry in a Pascal's triangle is identified by a row and a place. The rows are labeled starting from zero, so the first row would be row 0, the second one would be row 1, the third, row 2 and so forth. Places are given to each entry starting from the first number after 1, from the left to the right. 70 for example would be identified as row 8, place 4.
- The triangle is bordered by 1's on the right and left edges.
- The next line of numbers in diagonal order after the edge numbers are natural numbers 1,2,3,4...
- The next set of numbers inwards after the natural numbers are triangle numbers
- After the triangular numbers we have tetrahedral numbers in order 1,4,10,20...
- The next d-diagonal contains the next higher dimensional "d-simplex" numbers.
- The first number after 1 in each row divides all the other numbers in that row if and only if it is prime.
More patterns within Pascal's triangle
Pascal's triangle contains a number of smaller patterns within it. Some of these patterns include:
These are just a few of the patterns observed in Pascal's triangle. Other patterns include:
Animation Demonstrating Patterns
The Flash animation below allows you to explore some of the patterns present in Pascal's Triangle:
Applications of Pascal's Triangle
Heads and Tails
Pascal's triangle can be used to determine the combinations of heads and tails we can have depending on the number of tosses. From the possible outcomes, we can calculate the probability of any combination.
For example, if we toss a coin twice, we will have the combination HH once, TH twice and TT once, thus the possible outcomes would be in the order 1 2 1.This is also the same as the second row of Pascal's triangle. In general, if we toss a coin times, the combination of possible outcomes would be the horizontal entries in the th row of the Pascal's triangle.
|Tosses||Possible Outcomes||Pascals Triangle|
|1||H T||1 1|
|1 2 1|
HHT HTH THH HTT THT TTH HTT THT TTHTTT
|1 3 3 1|
What is the probability of getting exactly 2 heads with 3 coin tosses?
To answer this, we could use Pascal's triangle. The third row of Pascal's triangle has the entries 1 3 3 1=8 possible outcomes. Therefore we have 8 possible outcomes. 3 of the possible 8 outcomes give exactly 2 heads, therefore the probility of getting exactly 2 heads from 3 coin tosses is 3/8=37.5%.
Pascal's triangle can be used to determine binomial coefficients in binomial expansions. For example
In the above example, notice that the coefficients in the expansion are exactly the same as the entries in row two in Pascal's triangle.
This is summarized in the binomial theorem which states that in general,
where are the entries in row in Pascal's triangle.
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