How do you find the binomial expansion of ${(x - 2 y)}^{6}$ $(x-2y)^6$ ?

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How do you find the binomial expansion of ${(x - 2 y)}^{6}$ $(x-2y)^6$ ?

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Answer 1 · 2015-06-06T10:57:28

Use Pascal's triangle and choose the row that starts with $1, 6$ $1, 6$ , that is:

$1, 6, 15, 20, 15, 6, 1$ $1, 6, 15, 20, 15, 6, 1$

Write out the powers of $2$ $2$ up to $2^{6}$ $2^6$ as a sequence:

$1, 2, 4, 8, 16, 32, 64$ $1, 2, 4, 8, 16, 32, 64$

Multiply the two sequences together:

$1, 12, 60, 320, 480, 192, 64$ $1, 12, 60, 320, 480, 192, 64$

Alternate the signs:

$1, - 12, 60, - 320, 480, - 192, 64$ $1, -12, 60, -320, 480, -192, 64$

Hence:

${(x - 2 y)}^{6}$ $(x - 2y)^6$

$= x^{6} - 12 x^{5} y + 60 x^{4} y^{2} - 320 x^{3} y^{3} + 480 x^{2} y^{4} - 192 x y^{5} + 64 y^{6}$ $=x^6-12x^5y+60x^4y^2-320x^3y^3+480x^2y^4-192xy^5+64y^6$

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How do you find the binomial expansion of ${(x - 2 y)}^{6}$ $(x-2y)^6$ ?

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