Binomial theorem (Milind Charakborty's question at Yahoo! Answers)

[Fernando Revilla]

Here is the question:

I know that the last two terms of (a + b)^n = n.a.b^n-1 + b^n
What are the last three terms of the same?
Also
What are the last four terms of the same?

Here is a link to the question:

What are the last three and four terms of (a + b)^n? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Milind Charakborty,

According to the binomial theorem: $$(a+b)^n=\displaystyle\sum_{i=0}^n{}\displaystyle\binom{n}{k}a^{n-k}b^k=\displaystyle\binom{n}{0}a^{n}+\displaystyle\binom{n}{1}a^{n-1}b^{}+\displaystyle\binom{n}{2}a^{n-2}b^{2}+\ldots\\+\displaystyle\binom{n}{n-3}a^{3}b^{n-3}+\displaystyle\binom{n}{n-2}a^{2}b^{n-1}+\displaystyle\binom{n}{n-1}a^{}b^{n-1}+\displaystyle\binom{n}{n}b^{n}$$ Using $\displaystyle\binom{n}{p}=\displaystyle\binom{n}{n-p}$: $$(a+b)^n=\ldots+\displaystyle\binom{n}{3}a^{3}b^{n-3}+\displaystyle\binom{n}{2}a^{2}b^{n-1}+\displaystyle\binom{n}{1}a^{}b^{n-1}+\displaystyle\binom{n}{0}b^{n}\\=\ldots +\frac{n(n-1)(n-2)}{3!}a^{3}b^{n-3}+\frac{n(n-1)}{2!}a^{2}b^{n-2}+na^{}b^{n-1}+b^{n}$$