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
...
Click here to download the Microsoft Excel file containing the construction of Pascal's Triangle and some of its patterns


Pascal's Triangle is constructed with ones for the border of the triangle and the inside numbers being the sum of the two numbers closest to it in the above row.  For example, 1+2=2 and 1+6=7.

Some other important properties of Pascal's Triangle:

    Pascal's triangle gives the binomial coefficient (n choose k) as the (k+1)th number in the nth row.
Binomial coefficients (n choose k) are defined by n!/(k!(n-k)!) for any positive integers n and any integer k such that 0 is less than or equal to k which is less than or equal to n.  Binomial coefficients give the coefficients for the Binomial Theorem or the expansion of (a+b)n into a sum of powers.

For example:
(a+b)1=a+b                                      These coefficients give the first row of the triangle
(a+b)2=a2+2ab+b2                           These coefficients give the second row of the triangle
(a+b)3=a3+3a2b+3ab2+b3                These coefficients give the third row of the triangle
(a+b)4=a4+4a3b+6a2b2+4ab3+b4     These coefficients give the forth row of the triangle
 

    Pascal's Triangle also contains the numbers of the Fibonacci Sequence.  The Fibonacci Sequence is constructed of two ones, then the sum of two ones (1+1=2), then the sum of the third term and the second term (1+2=3), then two plus the forth term (3+2=5) and so on.

Fibonacci Seqence= 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,1 44, 233, 377, ...

 Click here to download a file containing how to find the Fibonacci Sequence in Pascal's Triangle.

Here are some links to other web sites exploring Pascal's Triangle: