Why do we need Pascal's triangle?

Answer 1

Pascals triangle is important because of how it relates to the binomial theorem and other areas of mathematics. The binomial theorem tells us that if we expand the equation (x+y)n the result will equal the sum of k from 0 to n of P(n,k)*xn-k*yk where P(n,k) is the kth number from the left on the nth row of Pascals triangle. This allows us to easily calculate the exponential of binomials without ever having to resort to expanding term by term.

In addition, the way that the triangle is constructed allows us to observe that P(n,k) is always equal to nCk or n choose k. While this may not seem important, you often need to calculate combinations in Statistics and Pascals Triangle provides one of the easiest ways to calculate a large number of combinations at once.