Expert: Sherry Wallin - 3/29/2010

Question
Hi,

We're doing a very large report on Pascal's Triangle and I know the Triangle is packed with so much interesting patterns but I don't think I've found them all and they are really really hard to explain. Could you please help me?

So here's Pascal's Triangle:
(imagine it centered)

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
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

What are the properties of this triangle and how do we explain these properties in terms of combinatorial coefficients?

Why are the numbers in the Pascal's triangle called binomial coefficients?

Thank you so much!

Answer
Mary~
    If you look at the first (top) row of PT (Pascal's Triangle), 1 is the  coefficient for (a+b)^0 [any number to the zero power is 1] and the second row is for the coefficients for (a+b)^1 [1*a+1*b] and the third row is for the coefficients for (a+b)^2 [1*a^2+2ab+1*b^2]. Notice the row number is always one more than the power of the binomial {first row is 0 power, 2nd row is first power, 3rd row is second power etc].
     In any given row you can add two adjacent numbers and they will sum to the number below the pair [witness in row 3 that 1+1 in the second row equals 2 in the third row.
     In any given row the coefficient for the binomial is nCr (that is, n choose r, a combination where n is the power of the binomial or it is the n+1 row). In the 4th row (so n+1 = 4 which means n = 3, i.e., (a+b)^3 and the coefficients are 1 3 3 1 and they correspond to 3C0,3C1,3C2,3C3 ->
where 3C0 is 3!/(3-0)!0!; 3C1 = 3!/(3-1)1! = 3!/2!1! = 3*2!/2!1! = 3/1 = 3 and likewise 3C2 = 3!/(3-2)!2! = 3!/1!2! = 3*2!/1!2! = 3/1 = 3 and finally 3C3 = 3!/(3-3)!3!= 3!/0!3!= 3!/1*3! = 1. Suppose you wanted the coefficients for the 6th row, that is, the binomial with a power of 5 then the coefficients would be 5C0 = 1,5C1 = 5,5C2 = 10,5C3 = 10,5C4 = 5, and 5C5 = 1.
     There are many others ways to view PT but these are the most common.

Math Prof