Question 1158319: Explain how to find the fifth term of the expansion of (2x+3y)^9 without writing out the entire sequence. Found 3 solutions by Boreal, MathLover1, greenestamps:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! From binomial expansion, the first part of the fifth term is 9C4. The (2x) is raised to the number of the term and the (3y) is raised to the complement (9-the number of the term, here 5). It looks like this below. The first term in the parentheses is raised to the progressively smaller power, and the second term Is raised to the progressively larger power.
9C0(2x)^9*3y^0
9C1(2x)^8*3y^1
9C2(2x)^7*(3y)^2
9C3(2x)^6*(3y)^3 and
9C4(2x)^5*(3y)^4, which is one way to write it
Picture the expression as 9 factors of (2x+3y) being multiplied together.
The first term in the product will use the x term from all 9 factors;
The second term will use the x term 8 times and the y term 1 time;
...
The fifth term will use the x term 5 times and the y term 4 times.
The first part in the calculation of the fifth term is determining the number of different ways you can choose 5 of the 9 factors to be the ones in which the x term is chosen. That number of ways is "9 choose 5":
In each of those 84 cases, the (2x) factor is used 5 times and the (3y) factor is used 4 times.
So the fifth term in the expansion is
=
Use a calculator if you need to multiply all that out.
