## Recreational Math## Just For Fun |

## How can one visualize Pascal’s triangle using algebra? |

One way of looking at Pascal’s triangle is through its connection to algebra. For example, expand (or remove the brackets around) the expression (1 + x)^{2} = (1 + x)(1 + x) = 1 + 2x + x^{2}. The same can be done with a cube; for example, (1 + x)^{3} = (1 + x)(1 + x)(1 + x) = (1 + x)(1 + 2x + x^{2}) = 1 + 3x + 3x^{2} + x^{3}; and even the expression (1 + x)^{4}, which equals 1 + 4x + 6x^{2} + 4x^{3} + x^{4}.

The coefficients (the numbers in front of the x’s) in the results are the connection. For the first example, the coefficients are 1, 2, 1; for the second one, 1, 3, 3, 1; and for the last expression, the coefficients are 1, 4, 6, 4, 1. These, of course, are the third, fourth, and fifth lines from Pascal’s triangle.