Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How does $ \binom{n}{k} $ 'n choose k' get involved with coefficient of $ (a+b)^n $. Is there any intuitive geometrical picture (interpretation) that it seems obvious?

share|improve this question
1  
This question seems to be related: Binomial Coefficients in the Binomial Theorem - Why Does It Work Question –  Martin Sleziak Jun 11 '12 at 13:03
    
Ah thanks ... !! i guess it's exactly what i'm looking for –  Santosh Linkha Jun 11 '12 at 13:04
3  
Picture an $n$-dimensional cube (that is the hard part) with side length $a + b$ and divide it up... this is not easy to do beyond $n = 3$ but it is a good exercise anyway. –  Qiaochu Yuan Jun 11 '12 at 14:07
    
How do I divide it?? .. i mean how many smaller cubes am i going to have?? –  Santosh Linkha Jun 11 '12 at 17:01
2  
There is a very good picture under "geometric explanation" in the Wikipedia article on the binomial theorem here: en.wikipedia.org/wiki/Binomial_theorem –  Jair Taylor Jun 14 '12 at 19:01

2 Answers 2

up vote 5 down vote accepted

Hint: Imagine writing $(a+b)^n$ as $(a+b)(a+b)\dots(a+b)$, and then multiplying out all the brackets. Ask yourself how many ways you can get a term involving $a^kb^{n-k}$.

share|improve this answer
    
thanks ... !! it's a lot more intuitive than comparing pascal's triangle –  Santosh Linkha Jun 11 '12 at 13:07

Expanding on what Old John wrote, it might help to consider a "noncommutative" version of the binomial theorem. $(a+b)^n = (a+b)(a+b)...(a+b)$ is going to have $2^n$ terms. Each of the $2^n$ words of length $n$ consisting of the letters $a$ and $b$ will occur exactly once. If you identify words via commutativity of multiplication, you will see there are $\binom{n}{k}$ words in the equivalence class of $a^{n-k}b^k$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.