Binomial Expansion?

Question

We write Binomial expansion of (1+x)^n as:
(1+x)^n = nC0+nC1x+nC2 x^2+...........+nCrx^r......+nCn...
Where sign "C" means "combination".My question is Why Combination is used or what is the importance of Combination in Binomial theorem? Why permutation is not used?Please tell me.

Answer 1

Combination is mathematically how many ways to choose a number of things from a set, counting different orders of the same chosen set as only one, whereas permutation counts them differently. For example, 3 choose 3 is = 1, because if you have three things and you choose 3, you get all of them only one way. 3P3 is 6, because you could have 123, 132, 213, 231, 312, 321.

Now, to answer your question: Let's say you want to expand (1+x)^3. This is =(1+x)*(1+x)*(1+x). The expansion by the binomial theorem is 3c0 + (3c1)x + (3c2)x^2 + (3c3)x^3. Look at the first term of that, 3c0 with no variable (that is, x^0). This is because there are 3 X's in the multiplication but you are choosing 0 of them. To obtain the x^1 term, you choose 1 of the 3 X's, etc. This is choosing because order does not matter in multiplication, 1*1*x = 1*x*1 = x*1*1, therefore you use nCr the combination rather than permutation

Answer 2

Combination is used because the Binomial Theorem uses two numbers to solve C0, C1, C2 etc.

Combination denotes that C derives from the elements of n and k of the Binomial Theorem.

Answer 3

It comes from Pascal's triangle. See the link below.
http://en.wikipedia.org/wiki/Binomial_coefficient

Answer 4

purmutations means no of orders or arrangements possible,
while cominations means simple combination