| PREVIOUS PAGE | COVER PAGE | TABLE OF CONTENTS | INDEX | PROBLEMS FOR THIS SECTION | NEXT PAGE |
We now tabulate the coefficients of (a + b)n
for n = 0, 1, 2, 3, 4, ... in a triangular array:
n Coefficients of (a + b)n 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 ... . . . . . .One may observe that the array is bordered with 1's and that each number inside the border is the sum of the two closest numbers on the previous line. This observaion simplifies the generation of additional lines of the array. For example, the coefficients for n = 5 are 1, 1 + 4 = 5, 4 + 6 = 10, 6 + 4 = 10, 4 + 1 = 5, and 1.
The above triangular array is called the Pascal Triangle in honor of the mathematician Blaise Pascal (1623-1662). A notation for the coefficients of (a + b)n is
For example, one writes (a + b)4 in this notation as
where
A two-term expression is called a binomial, and an expansion
for an expression such as (a + b)n is called
a binomial expansion. The coefficients listed in (6)
above are called binomial coefficients.
| PREVIOUS PAGE | COVER PAGE | TABLE OF CONTENTS | INDEX | PROBLEMS FOR THIS SECTION | NEXT PAGE |