[lorijessica16]

In the binomial expansion of (a+b)^12 determine

A) the value of k in the term containing a^2k b^k
B) the coefficient of that term, first in c(n,r) form, and then evaluate
C) the number of terms in the expansion
D) the middle term

I don't understand
This is from my statistics/ data management class thanks

[Perito]

This looks to be a pretty complete explanation of what's involved. Look it up and if you still have problems, post back.

Binomial theorem - Wikipedia, the free encyclopedia

[lisanoce]

I understand the expansion I just don't get how to solve this problem because it looks to be different

[Perito]

It's not different.

You can do it the brute-force way using Pascal's triangle. Keep going until the second number = 12.
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

But it gets a bit long at 12.

You can use equation 1 on the Wikipedia page:

Binomial theorem - Wikipedia, the free encyclopedia

to derive any term including the coefficient. That's all you're asked to do.

[Unknown008]

For (C); Or you could try seeing the pattern, quadratics have 3 terms

(a+b)^2 = a^2 + 2ab + b^2

Cubics four terms,

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

[galactus]

Do you have a nice calculator? Just use that for an easy way out.

Expand

There is a distinct pattern to these. Let me show you a trick to finding the coefficients without all that combinatorics stuff. OK?

The first term is . Since , we do not bother writing it.

To find the next coefficient, it is always the power term.

So, subtract 1 from the 'a' power and add 1 to the 'b' power.

The next term is then

Now, to find the next coefficient, multiply 12 by the 'a' power and then divide by the 'b' power plus 1.

We get (12*11)/2=66

Therefore, the next term is

The next term would be (66*10)/3=220,

The next coefficient would be (220*9)/4=496

Continue in that manner until you get to a coefficient of 924, then they repeat back down to the end at b^12.

See?

How many terms are there? Well, one more than the power of given binomial. Wouldn't that be 13?