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  Pascal's Triangle  

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What is Pascal's Triangle?
How do you construct it?
What is it used for?

Pascal's Triangle is an arithmetical triangle you can use for some neat things in mathematics. Here's how you construct it:
                             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 start out with the top two rows: 1, and 1 1. Then to construct each entry in the next row, you look at the two entries above it (i.e. the one above it and to the right, and the one above it and to the left). At the beginning and the end of each row, when there's only one number above, put a 1. You might even think of this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add up the number above and to the left (since there is no number there, pretend it's zero) and the number above and to the right (1), and get a sum of 1.

When people talk about an entry in Pascal's Triangle, they usually give a row number and a place in that row, beginning with row zero and place zero. For instance, the number 20 appears in row 6, place 3.

That's how you construct Pascal's Triangle. Here's an interactive version where you can specify the number of rows you want to see and from which you can bring up a large version that goes through row 19.



Where do we use Pascal's Triangle?

Pascal's Triangle is more than just a big triangle of numbers. There are two major areas where Pascal's Triangle is used, in Algebra and in Probability / Combinatorics.

  Algebra

    Let's say you have the polynomial x+1, and you want to raise it to some powers, like 1,2,3,4,5,.... If you make a chart of what you get when you do these power-raisings, you'll get something like this:
   (x+1)^0 =                        1
   (x+1)^1 =                   1    +    x
   (x+1)^2 =              1    +   2x    +    x^2
   (x+1)^3 =          1   +   3x    +   3x^2  +    x^3
   (x+1)^4 =      1   +  4x    +   6x^2  +   4x^3  +    x^4
   (x+1)^5 =  1   +  5x   +  10x^2  +  10x^3  +   5x^4  +    x^5 .....

    If you just look at the coefficients of the polynomials that you get, you'll see Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are called the binomial coefficients.

    There's a pretty simple formula for figuring out the binomial coefficients:

                   n!
       [n:k] = --------
               k! (n-k)!
                              6 * 5 * 4 * 3 * 2 * 1
       For example, [6:3] =  ------------------------  =  20. 
                              3 * 2 * 1 * 3 * 2 * 1
    
  Probability/Combinatorics

    The other main area where Pascal's Triangle shows up is in Probability, where it can be used to find Combinations. Let's say you have five hats on a rack, and you want to know how many different ways you can pick two of them and wear them. It doesn't matter to you which hat is on top, it just matters which two hats you pick. So this problem amounts to the question "how many different ways can you pick two objects from a set of five objects?"

    The answer? It's the number in the second place in the fifth row, i.e. 10. (Remember that the first number in the row, 1, is always place 0.)

                                 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
    
    Because of this choosing property, the binomial coefficient [6:3] is usually read "six choose three." If you want to find out the probability of choosing one particular combination of two hats, then that probability is 1/10.

    In about 1654 Blaise Pascal started to investigate the chances of getting different values for rolls of the dice, and his discussions with Pierre de Fermat are usually considered to have laid the foundation for the theory of probability.

  Triangular Numbers, Fibonacci Numbers

    The triangular numbers and the Fibonacci numbers can be found in Pascal's triangle. The triangular numbers are easier to find: starting with the third one on the left side go down to your right and you get 1, 3, 6, 10, etc.
                                 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
    
    
    The Fibonacci numbers are harder to locate. To find them you need to go up at an angle: you're looking for 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 1+5+6+1.



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