Pascal's Triangle

The Pascal’s triangle is one of the most interesting patterns ever. It was created by the French mathematician and philosopher Blaise Pascal. In order to build the triangle, you have to start with the number 1at the top, and then continue placing numbers below it in a triangular pattern. Each number is the result of the two numbers above it added.

Pascal’s triangle is a number triangle with numbers arranged in staggered rows.

One of the patterns of Pascal’s Triangle is displayed when one finds the sums of the rows. In doing so, it can be established that the sum of the numbers in any row equals 2n, when n is the number of the row. For example:

1 = 1 = 20

1 + 1 = 2 = 21

1 + 2 + 1 = 4 = 22

1 + 3 + 3 + 1 = 8 = 23

1 + 4 + 6 + 4 + 1 = 16 = 24.

As one familiar with algebra may notice, the numbers in each row of the triangle are precisely the same numbers that are the coefficients of binomial expansions. For example, when one expands the binomial, (x + y)3, algebraically it equals 1x3 + 3x2y + 3xy2 + 1y3. The coefficients of this binomial expansion, 1 3 3 1, correspond exactly to the numbers in the third row of Pascal’s Triangle. In general, the nth row in Pascal’s Triangle gives the coefficients of (x + y)n.

There’s a formula , where is a binomial coefficient.

In this formula, the n is equal to the row number of the triangle, while the r is equal to the element number in that particular row.

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