Pascal's Triangle

Start with 1. Then write two 1s below that. Then add up the two 1s, and write a 2 below, preceded and followed by a 1. Keep going, adding two numbers on each row and writing the sum on the next row, and you will get this:

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

This is referred to as Pascal's triangle.

There are many interesting patterns in this triangle. For example, the numbers in the 2nd diagonal are the integers, the numbers in the 3rd diagonal are the triangular numbers (1 + 2 + ... + (n-1) + n), the numbers in the 4th diagonal are the tetrahedral numbers (the sums of the triangular numbers starting with 1).

Also, the sum of the n-th row is equal to 2^n-1.

Pascal's triangle also appears in combinatorics. The number of ways to choose N objects out of a set of K objects is the N-th number on the K-th row of Pascal's triangle not including the topmost 1. For example, the number of ways to choose 3 objects out of 5 is the 3rd number on row 5 of Pascal's triangle, or 10.

The numbers in Pascal's triangle appear in powers of numbers of the form 10ⁿ + 1, until they overlap:

11² = 121 101² = 10201

11³ = 1331 101³ = 1030301

11⁴ = 14641 101⁴ = 104060401

11⁵ = 161051 101⁵ = 10510100501

101⁶ = 1061520150601

101⁷ = 107213535210701

101⁸ = 10828567056280801

A similar triangle can be devised for the trinomial coefficients, by adding up the 3 numbers above instead of 2:

1

1 1 1

1 2 3 2 1

1 3 6 7 6 3 1

1 4 10 16 19 16 10 4 1

1 5 15 30 45 51 45 30 15 5 1