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In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle.
In simple terms, Pascal's triangle can be constructed in the following manner. On the first row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left (if any) and the number directly above and to the right (if any) to find the new value. For example, the numbers 1 and 3 in the fourth row are added to produce 4 in the fifth row. More formally, this construction is using Pascal's identity, which states that
for positive integers n and k where n ≥ k and with the initial condition
Pascal's triangle generalizes readily into higher dimensions. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron. A higher-dimensional analogue is generically called a " Pascal's simplex ". See also pyramid, tetrahedron, and simplex.
Here are 14 lines of Pascal's triangle
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
Pascal's triangle has many uses in binomial expansions. For example
Notice the coefficients are the third row of Pascal's Triangle - 1,2,1. In general, when a binomial is raised to a positive integer power we have:
where the coefficients ai in this expansion are precisely the numbers on the nth row of Pascal's triangle; in other words, .
Also, when a Pascal's Triangle is constructed with 2n levels and all odd numbers are shaded, the result is an approximation to the Sierpinski triangle.
Some simple patterns are immediately apparent in Pascal's triangle:
There are also more surprising, subtle patterns. From a single element of the triangle, a more shallow diagonal line can be formed by continually moving one element to the left, then one element to the top-left, or by going in the opposite direction. One such example is the line with elements 1,6,5,1, which starts from the row, 1,3,3,1 and ends three rows down. Such a "diagonal" has a sum that is a Fibonacci number. In our example's case, 13. Observe:
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1The second highlighted diagonal has a sum of 233.
In addition, the sum of the squares of the elements of the nth row equals the middle element of the 2nth. For example, . In general form, that is:
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