## Monday, January 14, 2013

### Pascal's Triangle

You may have seen Pascal's Triangle at some time in your education. It is an array of numbers where each row has one more number in it than the last row. The top row is just the number 1 and the second row is 1 and 1, so these rows look like this.

1
1  1

The rule for creating the entries of a row are based on the row just above it. Here is how to state it as a sentence.

An entry in Pascal's Triangle is the sum of the entry just above it in the previous row and the number just above it and to the left.  If one of these numbers doesn't exist, assume it is zero.

That means the next row down will be 1(+0), 1+1 and (0+)1 so the first three rows look like

1
1  1
1  2  1

Like the Fibonacci sequence, the numbers go on forever.  Here are the first nine rows, which is the last row before the biggest number is more than two digits long.

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

The numbers are known as the binomial coefficients and they turn out to be the answers to a remarkable number of different questions about how to count things. The field of counting things in math is called combinatorics. This because it sounds more impressive to say "I study combinatorics" instead of saying "I count things for a living."

All this week we will look at the numbers in Pascal's Triangle and discuss the history how these numbers were used.

Tomorrow: The standard way to identify numbers in the array.