How do you find the coefficient of #a^(n-1)*b# in the expansion of #(a+b)^n#?

1 Answer
Dec 3, 2016

Answer:

The coefficient is #n#

Explanation:

The binomial theorem tells us that:

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

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

The term in #a^(n-1).b# is the one for #k=1#, with coefficient:

#((n),(1)) = (n!)/((n-1)! 1!) = n#

Alternatively, consider the product:

#(a+b)^n = overbrace((a+b)(a+b)...(a+b))^"n times"#

If we multiplied out the right hand side, then the only way we can get terms which contribute to the coefficient of #a^(n-1) b# is by picking the left hand #a# term of #n-1# of the binomials and one right hand #b# term.

We can do that in precisely #n# different ways, each contributing #1# to the coefficient.