Essay on Combinations in Pascal's Triangle

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Combinations in Pascal’s Triangle

Pascal’s Triangle is a relatively simple picture to create, but the patterns that can be found within it are seemingly endless. Pascal’s Triangle is formed by adding the closest two numbers from the previous row to form the next number in the row directly below, starting with the number 1 at the very tip. This 1 is said to be in the zeroth row. After this you can imagine that the entire triangle is surrounded by 0s. This allows us to say that the next row (row one) is formed by adding 0+1 to equal 1 and 1+0 to equal 1 making the next row 1 1. The second row is formed using the same rule so 0+1=1, 1+1=2, and 1+0=1 making this row 1 2 1. This rule can continue on into infinity making the triangle infinitely long.
Pascal’s Triangle falls into many areas of mathematics, such as number theory, combinatorics and algebra. Throughout this paper, I will mostly be discussing how combinatorics are related to Pascal’s Triangle.
Yang Hui has been found to be the oldest user of Pascal’s Triangle. But it is Blaise Pascal who around the year 1654 was credited for his extensive work on the many patterns of this triangle. Because of this people began to call it Pascal’s Triangle.
To demonstrate how combinations of elements of a set are showing up in Pascal’s Triangle, I will use the model on page 4. I set up this model by first placing blank boxes
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on the entire left side of the triangle (column 3) to represent n things going into groups of 0. There is only one way to do this, so every cell with a blank box can be said to have one item in it. I am using a box because it does not have a specific “value” but it is more of a holding place for new elements. In cell (B, 4), I placed an “a” because this represe...

... middle of paper ...

... the new possibility for 3 things going 3 at a time: “ab” combined with “c” becomes “abc.”
Below the chart that we have been following is a form of the original Pascal’s triangle that corresponds to the chart above it. If you count up all the pieces in a particular cell, you will see that their sum is equal to the corresponding cell in the recreated Pascal’s triangle.
In conclusion I would like to say that this discussion was not designed to be a proof of why combinations exist but an explanation of how these patterns occur. As you think about how combinatorics show up in Pascal’s Triangle, keep in mind that this is just one of the many patterns that are concealed within this infinitely long mathematical triangle.

Works Cited
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Pascal’s Triangle. Jessica Kazimir. Montclair State University. 28 July 2005. .

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