Difference between revisions of "Pascal's Triangle2"
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*The triangle is bordered by 1's on the right and left edges. | *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 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 <balloon title="load:trianglenumbers">triangular numbers</balloon><span id="trianglenumbers" style="display:none">This sequence is generated from a pattern of dots which form a triangle as shown in the image [[Image:trianglenumbers.gif]]</span> | + | *The next set of numbers inwards after the natural numbers are[[Pascal's_Triangle#Animation_Demonstrating_Patterns| <balloon title="load:trianglenumbers">triangular numbers</balloon><span id="trianglenumbers" style="display:none">This sequence is generated from a pattern of dots which form a triangle as shown in the image [[Image:trianglenumbers.gif]]</span> |
− | *After the triangular numbers we have <balloon title="load:tetrahedralnumbers">tetrahedral numbers</balloon><span id="tetrahedralnumbers" style="display:none">To understand what we mean by tetrahedral numbers, look at the stack of marbles in the image. Let each layer correspond to h=1,2,3...from the top to the bottom. For h=1, we need 1 marble. For h=2, we need 4 marbles (one at the top and three below it). For h=3, we need 10 marbles et cetra. The numbers 1,4, 10...are in a sequence referred to as tetrahedral numbers, owing to the tetrahedron arrangement like that shown in the image of the marbles. [[Image:tetrahedral.jpg]]</span> in order 1,4,10,20... | + | *After the triangular numbers we have <balloon title="load:tetrahedralnumbers">tetrahedral numbers</balloon><span id="tetrahedralnumbers" style="display:none">To understand what we mean by tetrahedral numbers, look at the stack of marbles in the image. Let each layer correspond to h=1,2,3...from the top to the bottom. For h=1, we need 1 marble. For h=2, we need 4 marbles (one at the top and three below it). For h=3, we need 10 marbles et cetra. The numbers 1,4, 10...are in a sequence referred to as tetrahedral numbers, owing to the tetrahedron arrangement like that shown in the image of the marbles. [[Image:tetrahedral.jpg]]</span>]] in order 1,4,10,20... |
*The next d-diagonal contains the next higher dimensional "d-[http://en.wikipedia.org/wiki/Simplex simplex]" numbers. | *The next d-diagonal contains the next higher dimensional "d-[http://en.wikipedia.org/wiki/Simplex 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. | *The first number after 1 in each row divides all the other numbers in that row if and only if it is prime. |
Revision as of 15:22, 10 July 2009
Pascal's Triangle |
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Pascal's Triangle
- Pascal's Triangle
Contents
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
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[[Pascal's_Triangle#Animation_Demonstrating_Patterns| 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
The further we go down Pascal's triangle, the more complicated the procedure becomes.
Horizontal Sum
Fibonacci Sequence
Sierpenski Triangle
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 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 |
2 | HH
HT TH TT |
1 2 1 |
3 | HHH
HHT HTH THH |
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
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.
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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