Binomial theorem?

Question

I've been on wikipedia reading about binomial theorem...

http://en.wikipedia.org/wiki/Binomial_theorem

I don't understand the notation they use. What is n and k in those parentheses.

Thanks guys. ;)

I'm not talking about permutations.

Answer 1

I can't reproduce it here exactly but

(n) means the number of ways of selecting k items out of n
(k) and it equals n!/[k!(n-k)!]

So suppose you flip 4 coins. There are

(4) ways that 2 coins can come up heads
(2)

(4) = [4!/2!(4-2)!] = 4*3*2*1/(2*1*2*1) = 6
(2)

TTHH THHT THTH HTHT HHTT HTTH

.. 1 ........ 2 ........3 ...... 4 ....... 5 ........ 6

Answer 2

When you use the theorem to expand (x+y)^n , there will be
k+1 terms, and thus k+1 coefficients. ( look at the examples on the wiki page.). they can be numbered as the 0th, the 1st, the 2nd,.....through the nth coefficient.

the formula for the binomial coefficient ,"n and k arranged vertically in parenthesis", gives the value of the kth coefficient when you raise ( x+y) to the n'th power.

For example:
n=3, k=0........formula gives you 1
n=3, k=1........formula gives you 3
n=3, k=2........formula gives you 3
n=3, k=3........formula gives you 1

So the four coefficients in the expansion of (x+y)^3 are
1,3,3,1. This agrees with

(x+y)^3 = x^3 + 3x^2 y + 3 x y^2 + y^3

Answer 3

n is the exponent of (x+y)^n.

k is the index of the summation sign, starting with k=0 , incrementing k by 1, until k = n.

So there will be an (x^n)(y^0) term, an (x^(n-1))(y^1) term, etc.