What is the fifth term of the expansion of #(a + b)^8#?

1 Answer
Feb 25, 2016

Answer:

#70a^4 b^4#

Explanation:

The fifth term is the middle term of nine, with coefficient given by all the ways of choosing #4# items out of #8#, namely the ways of choosing #4# #a#'s out of #8# binomial factors.

#((8),(4)) a^4 b^4=(8!)/(4! 4!) a^4 b^4#

#=(8xx7xx6xx5)/(4xx3xx2xx1) a^4 b^4#

#=1680/24 a^4 b^4#

#= 70a^4b^4#

The coefficient #((8),(4)) = 70# can be picked out as the middle term of the row of Pascal's triangle that begins #1, 8,...#:

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In full, we have:

#(a+b)^8 = sum_(k=0)^8 ((8),(k)) a^(8-k) b^k#

#=a^8+8a^7b+28a^6b^2+56a^5b^3+70a^4b^4+56a^3b^5+28a^2b^6+8ab^7+b^8#

which is just a particular example of the general Binomial Theorem:

#(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k#

where #((n),(k)) = (n!)/((n-k)! k!)#