1. ## binomial theorem

can someone pls help me solve the following expansion?

a binomial expansion, (a+b)^n, has 15 terms altogether. what is the value of n?

this question is part of my combinatorics chapter assignment. i don't know where else to put it

2. Originally Posted by oryxncrake
can someone pls help me solve the following expansion?

a binomial expansion, (a+b)^n, has 15 terms altogether. what is the value of n?

this question is part of my combinatorics chapter assignment. i don't know where else to put it
Hello. We know that $\displaystyle (x+y)^n=\sum_{k=0}^{n}{n\choose k}x^{n-k}y^k$. We know that no two of these terms can be coalesced into a single term (since they all have differing powers of x and y). Thus the number of terms is $\displaystyle n+1$...............so

3. where does n+1 come from?

4. also, does the "15" go anywhere in the equation?

5. Originally Posted by oryxncrake
also, does the "15" go anywhere in the equation?
How many terms does $\displaystyle (a+b)^1$ have?

What about $\displaystyle (a+b)^2$?

$\displaystyle (a+b)^3$?

:
:

notice the pattern?

CB

6. Yes. A better way maybe just to notice what CaptainBlack said.

What I was getting at was that

$\displaystyle \left(x+y\right)^n=\sum_{\ell=0}^{n}{n\choose \ell}x^{n-\ell}y^\ell$

And noting that on the RHS no two terms can be coalesced into a single term we can deduce that $\displaystyle \left(x+y\right)^n$ has as many terms as $\displaystyle \sum_{\ell=0}^{n}x^{n-\ell}y^\ell$. But how many does that have?

Well it not only has the first term through the $\displaystyle n$th term (which is $\displaystyle n$ terms) but it has the $\displaystyle 0$th term. So the number of terms in $\displaystyle \sum_{\ell=0}^{n}x^{n-\ell}y^\ell$ is $\displaystyle n+1$. Consequently the number of terms in $\displaystyle \left(x+y\right)^n$ is $\displaystyle n+1$