Can anyone explain pascals triangle?

Question

Using it for binomial expansion

Show me how it's done using the example (a-2)^6

Answer 1

The rows in the triangle are constructed so that each element in a row is the sum of the two elements immediately above it. So the first 7 rows are:

........... 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

To compute the binomial expansion of (a+b)^n, first you write out all the terms of the form a^kb^(n-k), starting with k=0 and ending with k=n. So for n=6, we would have:

a^6 + a^5 b + a^4 b^2 + a^3 b^3 + a^2 b^4 + a b^5 + b^6

Now, you give each of those numbers coefficients corresponding to the nth row in the triangle (remembering that the top row, containing only 1, is considered the 0th row, not the first). So in this case, the 6th row is 1 6 15 20 15 6 1. So giving these coefficients to the corresponding terms in the expansion, we get:

a^6 + 6a^5 b + 15a^4 b^2 + 20a^3 b^3 + 15a^2 b^4 + 6a b^5 + b^6

And now all that is left is to input the actual values of a and b. In this case, b=-2, so (a-2)^6 is:

a^6 + 6a^5 * (-2)b + 15a^4 * (4) + 20a^3 * (-8) + 15a^2 * (16) + 6a * (-32) + 64

Which is:

a^6 - 12a^5 + 60a^4 - 160a^3 + 240a^2 - 192a + 64

And we are done.

Answer 2

Pascal's triangle is a scheme of powers of a+b.

(1)-------------1---1
(2)-----------1---2---1
(3)---------1---3---3---1
(4)-------1---4---6---4---1
(5)----1---5--10--10--5---1
(6)--1---6--15--20--15--6---1
(7)1---7--21--35--35--21--7 1
...

The numbers in this scheme start with 1 and end with 1,
in between you have the sum of numbers above

It means f.ex that
(x+y)^5 = 1 x^5 + 5 x^4y + 10 x^3y^2 + 5 xy^4 + 1 y^5
Mark:
1) The coefficients are from the scheme
(1 5 10 10 5 1).
2) The sum of exponents is 5, the first (of x)
decreasing, the last (of y) increasing.
Now (a-2)^6 =
1 a^6 + 6 a^5(-2)^1 + 15 a^4(-2)^2 + 20 a^3(-2)^3 +
15 a^2(-2)^4 + 6 a(-2)^5 + 1.

I leave the evaluation of those terms to you.