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.      R!      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:

• There is a row of "1"s down each side
• Each number inside the triangle number comes from adding together the two numbers above it.

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:

• At Pongo McWhiffy's burger bar there are seven items on the menu. Pongo slices open a soggy bun for you and you choose any three different items to go inside. How many different combinations could you have?
  You could work it out with a few fiddly sums, but amazingly enough you can read the answer straight off the triangle. As there are 7 items to choose from you look at the 7th line (the one that starts 1,7,21...). As you can choose any 3 different items, you count in from the edge starting with zero until you get to 3. So you work along the line counting 0,1,2,3 and you find the number you finish on is 35. This turns out to be the answer. If you can choose 3 different items from a list of seven, there are 35 different ways you can do it!
  
  
• Suppose you've got an empty egg box with spaces for 6 eggs. You have 4 eggs to put in it. How many ways can you do this? Look at line 6 and count along 0,1,2,3,4 and you'll find the number you finish on is 15. That's the answer! Why not get an egg box and try it yourself?

But now for...

AN AMAZING DISCOVERY!

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:

• N is the total number of items to choose from
• S is the number of items you're allowed to choose.

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

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

• Try colouring in all the numbers that divide by 3.
• Try colouring in all the numbers that divide by 5
• Try choosing other numbers. If your triangle is big enough you'll see that prime numbers make nice clear patterns, and other numbers make more complex 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?"

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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