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### Course: Algebra (all content)>Unit 10

Lesson 26: Understanding the binomial theorem

# Binomial expansion & combinatorics (old)

An old video of Sal explaining why we use the combinatorial formula for (n choose k) to expand binomial expressions. Created by Sal Khan.

## Want to join the conversation?

• i did not get the part of having 3 unique permutations and 3 identical combinations @.
what do both of them mean?thanks
• Good question, here Sal coloured the three a's and b's differently to make them distinct. Now, among the different arrangements, we've got aab, aba and baa. As we know, in permutations, order matters, so these three are 3 unique permutations. However, for combinations, order doesn't matter, so all these three correspond to the identical combination a^2 b
(1 vote)
• Forgive me for asking, what would you guys think I ought to look over if I'm having a slight bit of trouble fully understanding this video?
• Thanks Dylan. I sussed it a while ago now, i was having issues knowing where to start mechanically with all this but i seem to have clicked with the formula. thanks again
• at wouldnt it be 3 different combinations and not indentical combinations as it was mentioned as the color of b and a in each of the possibility of a^2b is different? if not then why is all the different permutations showed in the case of a^2b but not in a^3 or b^3?
• Yes, you're right. He made a minor mistake there.
(1 vote)
• how do you do binomial expansions when the powers are rational and not integer?
• what do we call (a+b+c)^n?
is there a formal way of expanding this term?
(1 vote)
• Yes there is, actually.
http://en.wikipedia.org/wiki/Trinomial_expansion

Let's expand (a + b + c)².
= (a + b + c)(a + b + c)
= a(a + b + c) + b(a+ b + c) + c(a + b + c)
= a² + ab + ac + ab + b² + bc + ac + bc + c²
= a² + b² + c² + 2ab + 2ac + 2bc

(a + b + c)^3
= (a + b + c)(a + b + c)²
= (a + b + c)(a² + b² + c² + 2ab + 2ac + 2bc)
= a^3 + ab² + ac² + 2a²b + 2a²c + 2abc
+ a²b + b^3 + bc² + 2ab² + 2abc + 2b²c
+ a²c + b²c + c^3 + 2abc + 2ac² + 2bc²
= a^3 + b^3 + c^3 + 3a²b + 3a²c + 3ab² + 3b²c + 3ac² + 3bc² + 6abc
• At shouldn't 3a^2b = (3 choose 1)3a^2b?
• So in cases of binomial expansion, we treat permutations and combinations as same things?? Because in expansion of (a+b)^3, there are 3 terms which are a*a*b = a^2b (a squared into b); they are essentially 3 permutations, but just one unique combination.
• Why is it a combination instead of a permutation, if aba, baa and aab are not the same thing? I know it has something to do with the a's and b's being interchangeable, but what exactly is going on there?
(1 vote)
• If you think about it, aba, baa and aab are the same thing.
Think of it like this, if a=3 and b=2, then aba=3*2*3=18, baa=2*3*3=18 and aab=3*3*2=18.
Because of this, the order of 'selection' does not matter and therefore it is a combination rather than a permutation.