Pascal's Triangle
(The following presentation was originally written for a middle school mathematics event. You may print a handout summarizing this presentation that was distributed at the event.)
Pascal's triangle is an array of natural numbers. The initial row is referred to as row 0, and the left-hand column is referred to as column 0.
The initial row contains a single "1", and each row has one more entry than the preceding row. Each entry is computed by adding the numbers "above" and "above-left". The entry \(C(5,3)\) in row 5, column 3 is 10, because:
\[C(5,3) = C(4,2) + C(4,3) = 6 + 4 = 10\]The \(C\) in \(C(n,r)\) is a mathematical notation for "how many ways can a combination of r elements be selected from a set of n elements?" This formula for computing entries is known as Pascal's identity. Rows 0 through 6 of Pascal's triangle are shown in the figure above.
As you can see, this addition will quickly get out of hand! For example, the entry \(C(25,13)\) in row 25 and column 13 is \( 5{,}200{,}300\).