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Pascal's triangle

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Toothpaste:
... and I was all set to do absolutely nothing till next Friday.

Collin Li:
It's not on the course. You don't need to know how to find the coefficient of x^4 for (3x+2)^7.

Toothpaste:
Rejoice!

(But I'll learn it since I don't want to be left out of maths world.)

Collin Li:

--- Quote from: "Toothpick" ---Rejoice!

(But I'll learn it since I don't want to be left out of maths world.)
--- End quote ---


It's pretty simple:


--- Code: ---   1 1
  1 2 1
 1 3 3 1
1 4 6 4 1
   ...
--- End code ---


Notice how the numbers from the row above add up together to give the number below? That is Pascal's triangle.

The binomial theorem links to this by saying that you can generate any number from this triangle using nCr, where:
n = the row, and
r+1 = the column
...on Pascal's triangle.

This links into polynomial expansion:

For example:
(x+a)^3 = x^3 + 3ax^2 + 3a^2x + a^3

Notice the coefficients? (1, 3, 3, 1) This is row 3 of Pascal's triangle, for expanding a polynomial of degree 3.

Now, I think you're clever enough to figure out how to find out the coefficient to x^4 for (2x+3)^7

Toothpaste:
Hah, thanks!

Hmm..
(2x+3)^7 ..

so the coefficient of x^4 is

(7 C 4) (2x)^4 3^(3)

15120?

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