# How do I use Pascal's triangle to expand the binomial (a-b)^6?

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30
May 12, 2015

The 7th row of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1, which are the absolute values of the coefficients you are looking for, but the signs will be alternating.

${\left(a - b\right)}^{6}$

$= {a}^{6} - 6 {a}^{5} b + 15 {a}^{4} {b}^{2} - 20 {a}^{3} {b}^{3} + 15 {a}^{2} {b}^{4} - 6 a {b}^{5} + {b}^{6}$

Compare with the 'positive' case

${\left(a + b\right)}^{6}$

$= {a}^{6} + 6 {a}^{5} b + 15 {a}^{4} {b}^{2} + 20 {a}^{3} {b}^{3} + 15 {a}^{2} {b}^{4} + 6 a {b}^{5} + {b}^{6}$

For the 'negative' case, we replace $b$ with $- b$ and notice that the signs follow the odd/even parity of the power of $b$, because ${\left(- b\right)}^{n} = {\left(- 1\right)}^{n} {b}^{n}$.

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