How do you expand $(3a +b)^4$ using Pascal’s Triangle?

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Answer 1 · 2015-08-06T19:52:05

Multiply the $5$ th row of Pascal's triangle by a list of descending powers of $3$ to find the coefficients of the expansion:

$(3a+b)^4 = 81a^4+108a^3b+54a^2b^2+12ab^3+b^4$

Write down the $5$ th row of Pascal's triangle as a sequence:

$1$ , $4$ , $6$ , $4$ , $1$

Write down descending powers of $3$ from $3^4$ to $3^0$ as a sequence:

$81$ , $27$ , $9$ , $3$ , $1$

Multiply the two sequences together to get:

$81$ , $108$ , $54$ , $12$ , $1$

These are the coefficients of the terms in $a^4$ , $a^3b$ , ... , $b^4$ :

$(3a+b)^4 = 81a^4+108a^3b+54a^2b^2+12ab^3+b^4$

How do you expand (3a +b)^4 (3a +b)^4 using Pascal’s Triangle?