How do you expand # (x + y)^6# using Pascal’s Triangle?

1 Answer
Sep 1, 2015

The row of Pascal's triangle starting #1#, #6# gives the sequence of coefficients for the binomial expansion.

Explanation:

Write out Pascal's triangle as far as the row that begins #1#, #6# ...
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These are the coefficients you need for the expansion:

#(x+y)^6 = x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6#

Why does this work?

If we write out the value as a product of binomials we have:

#(x+y)^6 = (x+y)(x+y)(x+y)(x+y)(x+y)(x+y)#

If you pick one term from each binomial and multiply them together, then you have made #6# choices of left or right. If you choose all left, then you end up with #x^6# - which you can do just one way. If you choose all right then you end up with #y^6#, which again you can only do one way.

Otherwise, you are making a mixture of left and right choices, analogous to picking your way down from the top of Pascal's triangle to the bottom, via a sequence of left and right branches. The power of #x# resulting is the number of left branches you choose and the power of #y# the number of right branches.

Each number in Pascal's triangle is the sum of the two above, each of which counts the number of ways to reach that point by a sequence of left and right choices. So all of the numbers in Pascal's triangle count the number of ways to reach them by left/right choices starting at the top.