WOMAN: Exercise 8.
Find the middle term of (2x minus 3y) all to the power of 6.
The expansion of (2x minus 3y) all to the power of 6 has seven terms,
so the middle term is the fourth term.
The general term in the expansion of (2x minus 3y) all to the power of 6
is 6 choose r times 2x to the power of six minus r
times negative 3y all to the power of r,
where r is between 0 and 6 inclusive.
Given that the first term is the one in which r is equal to 0,
the fourth term is therefore the one in which r is equal to 3,
so the middle term is the term where r = 3.
So substituting in we get six choose three
times 2x all to the power of three
times negative 3y all to the power of 3,
which simplifies to give
20 times 8x cubed times negative 27y cubed,
which simplifies to give a middle term
of negative 4,320x cubed y cubed.