How do you expand $(1+2x)^6$ using Pascal’s Triangle?

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Answer 1 · 2016-01-17T22:35:03

Use the appropriate row of Pascal's triangle and a sequence of powers of $2$ to find:

$(1+2x)^6 = 1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6$

Write out Pascal's triangle as far as the row which begins $1, 6$ ...

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This gives you the sequence of coefficients for $(a+b)^6$ :

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

Then we can account for the factor of $2$ of the $2x$ term, by multiplying by a sequence of powers of $2$ :

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

to get:

$1, 12, 60, 160, 240, 192, 64$

Hence:

$(1+2x)^6 = 1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6$

How do you expand (1+2x)^6(1+2x)^6 using Pascal’s Triangle?