Algebra Pascal's Triangle row help!?

In Pascal's triangle each number, but the first and the last of each row, is the sum of the two numbers "above" as shown in the picture

to calculate 8th power you must build the triangle from row 0 to row 8 to get the coefficients of (x + y)^8

to get the letters you write a 8 degree polynomial which contains ONLY terms having degree 8, like x^3y^5

starting from x^8 then x^7 y then x^6 y^2 and so on until y^8

Then you apply each coefficient of the Pascal's triangle 8th row to the terms of the polynomial above and you're done

the resulting polynomial will have 8 + 1 terms

to verify if you understood well, try to work out

(a + b)^10 and verify if you find

a^10 + 10 a^9 b + 45 a^8 b^2 + 120 a^7 b^3 + 210 a^6 b^4 +

252 a^5 b^5 + 210 a^4 b^6 + 120 a^3 b^7 + 45 a^2 b^8 +

10 a b^9 + b^10

There's a basic pattern you can follow to fill in a Pascal's triangle.

Starting with (x+y)^0 you have:

1

Then with (x+y)^1, you have

1x + 1y

Then with (x+y)^2, you have

1x² + 2xy + 1y²

Hopefully you see a pattern, but if not, let me spell it out.

First, let's worry about the x and y terms. They will follow a pattern. The first term will be x^n. For example with (x+y)^8, it will be x^8. Then you just subtract 1 from the exponent on x and add it for y.

So the next term will be x^(n-1)y --> x^7y. The next term is x^6y², etc. until you get to y^8.

As for the coefficients, they are just the numbers you would put in a Pascal's triangle. The first and last coefficients are always 1 and then the rest are the sum of the two coefficients above.

............ 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

There is also a formula for them.

C(n,k) = n! / (n-k)! k!

So for example for (x+y)^8, you have C(8,0), C(8,1), ..., C(8,8)

That would be:

C(8,0) = 8! / 8! 0! = 1

C(8,1) = 8! / 7! 1! = 8

C(8,2) = 8! / 6! 2! = 28

C(8,3) = 8! / 5! 3! = 56

C(8,4) = 8! / 4! 4! = 70

C(8,5) = same as C(8,3)

C(8,6) = same as C(8,2)

C(8,7) = same as C(8,1)

C(8,8) = same as C(8,0)

So for the (x+y)^8 the final result would be:

1x^8 + 8x^7y + 28x^6y² + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x²y6 + 8xy^7 + 1y^8