The Fabulous Pascal's Triangle

Here's how Pascal's Triangle starts:

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

1      9      36     84      126      126     84     36      9      1

1      10     45    120     210      252     210    120     45     10      1

If you want to know any number on the triangle without writing it all out, you can use the "combinations" formula.
N! x (R-N)!
R = the row you want (this is indicated by the second number on the row, e.g. the sixth row is the one that starts 1-6-15...)
N = the number along the row. (You count along starting with 0. So if you didn't know the number 20 on the sixth row and wanted to work it out, you count along 0,1,2 and find your missing number is the third number.)
So to work out the 3rd number on the sixth row, R=6 and N=3. You work out R! =6x5x4x3x2x1 =720. N! = 3x2x1=6. (R-N)! = (6-3)! =3! =3x2x1 =6. So putting these into the formula we get 720/(6 x 6) = 20. As you can see, the third number on row 6 is 20 so the formula works!

Find out how to get
The Fibonacci Series
from Pascal's Triangle.
Qiu Zhe from China tells us that they call this triangle the JIAXIAN TRIANGLE after the Chinese mathematician Jiaxian who was working on it hundreds of years before Pascal!

It's dead simple to draw:

If you set out your own triangle, you can go on for ever. In Do You Feel Lucky? it tells you some of the amazing things this triangle can tell you. Here are a couple of examples:

But now for...


Whenever people talk about combinations, they always say that you have to choose different things - in other words when you choose your 6 lottery numbers they all have to be different, and when we first visited Pongo's we had to choose 3 different items off the menu. However, suppose the things you choose don't have to be different?

Now Pongo lets you choose 3 items off his menu - but they don't all have to be different! How many combinations are there?

As well as the combinations you could have before such as egg/burger/onions, or spouts/tomato/burger, you could also have things like egg/sprouts/egg or even sausage/sausage/sausage.

Where it gets confusing is that if you think that egg/sprouts/egg is different from sprouts/egg/egg, then the sums are simple. The number of different combinations are 7x7x7= 343. However if you think that egg/sprouts/egg is the same as sprouts/egg/egg, we need a fancier formula that elimates combinations which are the same...

To find out you could put the numbers into this formula:

OR.... look at Pascal's triangle again!

As there are 7 items to choose from, find the number 7, but this time look down the diagonal. (We've coloured it red in the triangle above.) As we can choose 3 items that don't have to be different, you count 0,1,2,3 down the diagonal and you'll find that there are now 84 possible combinations! If Pongo lets you choose 4 items that don't have to be different there are 210 combinations. Yuk.

As far as we know, this is the only page on the web showing this formula and how it fits with Pascal's triangle and that's why this page has a little copyright note at the bottom. We call it THE UNKNOWN FORMULA and it's now featured in The Perfect Sausage and Other Fundamental Formulas. If you want to know where it comes from then try The Unknown Formula Explanation.

Colouring in Pascal's Triangle

Go THIS way Go THAT way

If you draw out a big Pascal's triangle, it can make some amazing patterns.

If you don't want to draw out your own triangle, go to Dolly's Links and try the The Self-Colouring Pascal's Triangle. It's brilliant!

Back to "Do You Feel Lucky?"

Murderous Maths Main Index Page

This page dealing with the "non different" combinations formula and its association with Pascal's Triangle is copyright © Kjartan Poskitt 2000.
Any enquiries with respect to this should be addressed to
Kjartan Poskitt c/o Scholastic Books, Euston House, 24 Eversholt St, London NW1 1DB