Pascal's Triangle2

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Pascal's Triangle
Pascal2.gif
Field: Algebra
Image Created By: Math Forum
Website: Mathforum.org

Pascal's Triangle

Pascal's Triangle


Basic Description

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

Hockey stick pattern http://ptri1.tripod.com/

Look at the numbers highlighted in red in the image on the left. We can see that the sum of the ones in a line equals the entry not in a line. 1+6+21+56=84. This pattern holds for the other "hockey stick" selections.

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.


Properties

  • 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 triangular 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:

Magic 11's

For the first five rows of the Pascal's triangle, there's an interesting pattern with the number 11. If a row is made into a single number (omitting the spaces), the resulting number is equal to 11 raised to the n^th power, where n is the row number. For example:

Row # Actual Row Single Number Formula
0 1 1 11^0
1 1 1 11 11^1
2 1 2 1 121 11^2
3 1 3 3 1 1,331 11^3
4 1 4 6 4 1 14,641 11^4

After the fifth row, this pattern gets more complicated.

Horizontal Sum

If you add the numbers in each row horizontally, then look at the pattern formed by the resultant numbers, you will notice that th numbers double each time, but all are in the power of 2.

Fibonacci Sequence

The Fibonacci sequence is a list of Fibonacci numbers. Each Fibonacci number is generated by adding the previous two consecutive terms in the sequence. The first two Fibonacci numbers are 0 and 1. This pattern can be located in Pascal's triangle. The sum of the entries in each row, as shown in the animation demonstrating patterns, gives a Fibonacci number.

The numbers that result from summing the rows from top to bottom form the Fibonacci sequence.

Sierpenski Triangle

Sierpenski triangle generated by coloring odd and even numbers with Pascal's triangle with two different colors http://en.wikipedia.org/wiki/Pascal's_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. You could try this out in the animation and see what happens.

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 could have the any of the following combinations: 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 x times, the combination of possible outcomes would be the horizontal entries in the xth row of the Pascal's triangle.

Tosses Possible Outcomes Pascals Triangle
1 H T 1 1
2 HH

HT TH

TT
1 2 1
3 HHH

HHT HTH THH
HTT THT TTH

TTT
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%.


In Algebra

Pascal's triangle can be used to determine binomial coefficients in binomial expansions. For example

(x + y)^2 = x^2 + 2xy + y^2 =1{x^2}{y^0} + {2x^1}{y^1} +{1x^0}{y^2}

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,

(x + y)^n = {a_0}{x^n} + {a_1}{x^{n-1}} + {a_2}{x^{n-2}}{y^2}++ {a_{n-1}}{xy^{n-1}} + {a_n}{y^n}


where a_i are the entries in row n in Pascal's triangle.

In Combinatorics

References

Hockey stick pattern http://ptri1.tripod.com/
Triangular numbers http://www.mathsisfun.com/numberpatterns.html#triangular
Tetrahedral numbers http://www.mathsisfun.com/tetrahedral-number.html
Horizontal sum http://www.mathsisfun.com/pascals-triangle.html
Fibonacci sequence http://mathforum.org/dr.math/faq/faq.pascal.triangle.html
Sierpenski Triangle http://en.wikipedia.org/wiki/Pascal's_triangle




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