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At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both
formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the
Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3;
1+0=1. In this way, the rows of the triangle go on infinitely. A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in
that row. For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and
the last 1 is the 3rd element. The formula for nCr is:
n!--------r!(n-r)!
The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody figured out a formula for this expansion, and we can plug the binomial 3x – 2 and the power 10 into that formula to get that expanded (multiplied-out) form.
The Binomial Theorem
Expand (a + b)5
5 0 5 1 1 5 2 25 0 5 1 5 2
5 3 3 5 4 4 5 5 55 3 5 4 5 5
C a b C a b C a b
C a b C a b C a b
5 0 4 1 3 2 2 3
1 4 0 5
1 5 10 10
5 1
a b a b a b a b
a b a b
5 4 1 3 2 2 3 4 55 10 10 5a a b a b a b ab b
Expand (a -b)5
5 0 4 1 3 2 2 3
1 4 0 5
1 5 10 10
5 1
a b a b a b a b
a b a b
5 4 1 3 2 2 3 4 55 10 10 5a a b a b a b ab b
5 0 5 1 1 5 2 25 0 5 1 5 2
5 3 3 5 4 4 5 5 55 3 5 4 5 5
C a b C a b C a b
C a b C a b C a b
Expand (2a + 1)5
5 0 4 1 3 2 2 3
1 4 0 5
1 2 1 5 2 1 10 2 1 10 2 1
5 2 1 1 2 1
a a a a
a a
5 4 3 232 80 80 40 10 1a a a a a
5 0 5 1 1 5 2 25 0 5 1 5 2
5 3 3 5 4 4 5 5 55 3 5 4 5 5
C a b C a b C a b
C a b C a b C a b
Expand (3a-2b2)5
0 1 2 35 4 3 22 2 2 2
4 51 02 2
1 3 2 5 3 2 10 3 2 10 3 2
5 3 2 1 3 2
a b a b a b a b
a b a b
5 4 2 3 4 2 6 8 10243 810 1080 720 240 32a a b a b a b ab b
5 0 5 1 1 5 2 25 0 5 1 5 2
5 3 3 5 4 4 5 5 55 3 5 4 5 5
C a b C a b C a b
C a b C a b C a b
Expand (x2 + 3)6
6 0 6 1 1 6 2 26 0 6 1 6 2
6 3 3 6 4 4 6 5 5 0 66 3 6 4 6 5 6 6
C a b C a b C a b
C a b C a b C a b C a b
6 5 4 30 1 2 32 2 2 2
2 1 04 5 62 2 2
1 3 6 3 15 3 20 3
15 3 6 3 1 3
x x x x
x x x
12 10 8 6 4 218 135 540 1215 1458 729x x x x x x