Pascal's Triangle2
Pascal's Triangle |
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Pascal's Triangle
- Pascal's Triangle
Contents
Basic Description
Pascal's triangle is a triangular arrangement of certain numbers that show interesting patterns. We start out with 1 on the top row, which we will label as row 0. Every entry from the next row is the sum of the two numbers above it. If a number is absent on the diagonal left or the right, replace that empty entry with 0 and find the sum. We can repeat this process endlessly.
For instance, let's look at a row . This row can be seen as . We can find the entries for the next row by following the method above. The next row will be .
Pascal's triangle was first introduced in Pascal's paper Treatise on the Arithmetic Triangle published posthumously in 1665. In his original paper, the the triangle was rotated 45° so that each entry is the sum of the two preceding entries on the horizontal row and the vertical column, as shown in Image 1.
A More Mathematical Explanation
Constructing Pascal's Triangle
There is a mathematical rule to constructing Pascal's triangle. [...]Constructing Pascal's Triangle
There is a mathematical rule to constructing Pascal's triangle. This general rule is not necessary for the readers' understanding of the following sections, but it is helpful for the readers' understanding of the proofs of certain sections.
Let be a sequence for some Then we can
Let be a sequence for some . Then we can generate a new sequence that is in Pascal's relations with the original sequence according to the following rule:
as shown in Image 2.
For instance, if we have a sequence , we can produce a new sequence by following the rule above and get the sequence .
Now, we can construct Pascal's triangle starting from a sequence that consists of one term, 1. We can use Pascal's relations to generate further sequences and place these new sequence below the original sequence.
Before we proceed, we will define the terms rows and places that are used in Pascal's Triangle. The rows proceed as row 0, row 1, row 2, ..., as we start from the top. Places are given to each entry starting from the first number after 1, from the left to the right. 6, for example, would be identified as row 4, place 3.
We will use the notation:
for the entry that is on the th row and th place.
Patterns within Pascal's Triangle
Fibonacci Sequence
The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... where the first two numbers are 1s and every later number is the sum of the two previous numbers. The Fibonacci numbers can also be found in Pascal's triangle by adding the entries of diagonals as shown in image 4. You can try this at animation demonstrating patterns.
An easier way to show the Fibonacci numbers is to list the numbers on the diagonals on a triangular arrangement. Then, we get Image 5, and adding all the numbers in the same row gives the Fibonacci Sequence.
Sierpinski Triangle
If we are to color the odd and even numbers in Pascal's triangle with two distinct colors, we would observe an interesting recursive pattern seen in the Sierpinski triangle.
We can construct a Sierpinski triangle according to the following rule :
(1) Draw an equilateral triangle, as shown in the first step of Image 6.
(2) Then, connect the midpoints of each side of the triangle and remove the triangle in the center, as shown in step 2 of Image 6.
(3) For the remaining three black triangles, repeat step 2.
(4) Repeat step 2 for any remaining triangles.
In fact, if we repeat this process until the number of rows in the triangle approaches infinity, we get Sierpinski's triangle. You could try this out in the animation and see what happens.
Hockey Stick Pattern
Horizontal Sum
Other Patterns and Properties
- The triangle is symmetrical. The numbers on the left side of the triangle have identical numbers on the right side. In other words, the th number from the left at any row is the same as the th number from the right at the same row; thus, . For instance, look at the 5th row. 10, which is the 3rd from the left, is also 3rd from the right in the 5th row.
- 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
- The next set of numbers inwards after the natural numbers are triangular numbers that continues as
- After the triangular numbers we have tetrahedral numbers that continue as .
- 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.
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
We can use Pascal's triangle to find the probability of having a certain outcome of tossing coins.
For example, if we toss a coin twice, we could have the any of the following results: Head-Head once, Tail-Head twice and Tail-Tail once. (Here, we do not differentiate the coins. Thus, Tail-Head and Head-Tail are considered the same combination.) The number of each possible outcome is 1, 2, 1, which is also the same as the second row of Pascal's triangle.
In general, if we toss a coin times, the number of each possible outcome is the same as the horizontal entries in the th row of the Pascal's triangle.
What is the probability of getting exactly 2 heads with 3 coin tosses?
To answer this, we could use Pascal's triangle. Adding the entries of the third row of Pascal's triangle, we get 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 .
Binomial Coefficients in Pascal's Triangle
Pascal's triangle can be used to determine binomial coefficients in binomial expansions. For example, when we consider the expansion of for we get:
- .
Notice that the coefficients in the expansion are the entries in the th row in Pascal's triangle. In general, we can use binomial coefficients to describe the coefficient of the expansion as :
- .
This is called the binomial theorem. Indeed, the binomial coefficient is the entry of the row and th place in Pascal's triangle, and we can find the value by:
We know that
- .
Now, let's consider the expansion of .
This can be written as :We know that
- .
Now, let's consider the expansion of .
can be written as :
We can find the binomial expansion of by expanding the right side and left side of Eq. (2), and the expansion of either side of the equation must give the same result.
Expanding the left side of Eq. (2), we get:
Expanding the right side of Eq. (2), we get:
The left side expansion and right side expansion must be the same. Thus, the coefficients for terms with same powers of and must also be the same. Thus,
This is the same form as Pascal's relations in Eq. (1). We can see that the sequence of coefficients of the expansion of is in Pascal's relations with the sequence of coefficients of the expansion of . The very first sequence for is :
- ,
which is the sequence in the th row of Pascal's triangle. Thus, all other sequences of coefficients of binomial expansion also coincide with the entries of Pascal's triangle.
Combinations in Pascal's Triangle
Let be a set of elements. Then, we can find a subset of that contains different elements. This subset is called a k-combination. The number of k-combination of an element set is equal to the number of things taken at a time. This number is denoted by , , or , and is equal to the binomial coefficient:
Thus, we can find the number of k-combinations from Pascal's triangle.
Teaching Materials
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References
Uspenskii, V. A. (1974) Pascal's Triangle. Chicago : The University of Chicago Press.
Posamentier, Alfred S & Lehmann Ingmar. (2007) The Fabulous Fibonacci Numbers. New York : Prometheus Books.
Wikipedia (Combination). (n.d.). Combination. Retrieved from http://en.wikipedia.org/wiki/Combination.
Wikipedia (Pascal's Triangle). (n.d.). Pascal's Triangle. Retrieved from http://en.wikipedia.org/wiki/Pascal's_triangle.
Future Directions for this Page
- New Definition of Pascal's Triangle in the more mathematical explanation
-Current definition is based on Pascal's relations given by the sequence . Maybe change the definition using Combinations. Pascal's triangle is an arrangement of binomial coefficients. For instance, the 0th row is , the 1st row is and so forth. -With the new definition, come up with new proofs for the properties
- New Structure/ Organization
-In the think alouds, people were more interested in the application of Pascal's triangle (ex, the coin toss) more than to the properties and patterns within Pascal's triangle. Maybe think about moving the application section to the top.
- number Images, anchor them
- rewrite the coin-toss section to make everything clear
- Add an why interesting section, if possible
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