First Encounter With Math: Pascal's Triangle

By Sebastiaan

Thinking back, the first time I fell in love with mathematics must have been reading the magnificent book "De Telduivel", the dutch translation of "The Number Devil" by H.M. Enzensberger.

The part I found most fascinating, was what I later learned is called "Pascal's Triangle" named after Blaise Pascal, a famous western mathematician, physicist and religious philosopher.

First let me explain a bit about the triangle. "1" is our top number, which comes in the uppermost 'corner' of the triangle. Then we go downwards and left, where we put another "1" and we go downwards right, where we put a third "1". Let me sketch the situation for you:

1

1   1

Now, we go down-left, and down-right from both ones. The number we put in the space, is the sum of both numbers above it, and in the leftmost and rightmost spaces we just put a "1". Continuing this process infinitely, we get Pascal's Triangle:

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

You get the general idea.

Now, the most interesting thing about the triangle, is that it has a wide variety of uses. (I understand that almost everyone who reads this will lose interest in a few moments, but I still would like to share this)

One practical use is found in the calculation of combinations. Say, we are looking for the number of combination of n things taken k at a time. Mathematicians call this n choose k. Let's take a practical example, which we find in the game of poker.

Take a look at my example: we have n = 52, for there are 52 cards in a deck of cards, and let's say k = 5 because we have a poker hand with 5 cards. We now have 52 choose 5. The official formula for combinations is n! divided by k! times (n-k)!, so let's put that into perspective...

...which means there are 2,598,960 combinations of poker hands.

Could we not have found this answer in a differnet way? The answer is yes. Take a look at the following -much simpler- example: 3 choose 1. After a few calculations we find the answer is the simple "3". Take a look at the 3rd row of Pascal's Triangle (don't count the top "1" as a row, this can be seen as the starting number, or row 0), and then the first number that isn't one (for that would be 3 choose 0). This is also 3!

This works for all combinations. 7 choose 3 would be 35, according to the triangle. According to the formula this is 7! (so 7 times 6 times 5 times 4 ..... times 1) divided by 3!4!:

(7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1 x 4 x 3 x 2 x 1)

...which is indeed 35.

As a kid, I used to write down the triangle as far as I could, which mostly meant 'till I ran out of paper... I especially used to do this during the more boring classes in primary school. It's ages ago, though I remember it as if it were yesterday.

I will probably post more about the triangle later on, since I so dearly love it. There are some great patterns in it...