Sample spaces having equally likely outcomes

[blob84]

If we flip a coin n-times,
what is the probability of the event $$A= \left \{there \space are \space k \space head \right \}$$.
I should find the number of elements of A,
the book says that is $$\binom{n}{k}$$ but for $$n=3$$ and $$k=2$$, all the possible outcomes are:
$$A= \left \{(h, h,h), (h, h, x), (h, x, h), (x, h, h) \right \}$$, where the position of h or x is the k-flip.
How to find this number?

PS. h is head.

 

[tiny-tim]

hi blob84!

i'm sorry, i don't understand this at all

your example seems to be k = 3, not n = 3, and i don't understand what those four outcomes are

can you explain again?​

 

[blob84]

k is the number of the head in A, int the example k = 2, any vector of A has at least two head.
you flip a coin n-times, so if n = 3 you flip the coin 3 times, the problem is to count the number of vectors in A.

PS. h is head.

 

[tiny-tim]

int the example k = 2, any vector of A has at least two head.

ah, now i see what you meant

no, if k = 2, there must be exactly 2 heads

so the possible outcomes are

1: xxx k = 0 (0 heads)

3: xxh xhx hxx k = 1 (1 head)

3: xhh hxh hhx k = 2 (2 heads)

1: hhh k = 3 (3 heads)​

 

[blob84]

yes only 2 head, oh my god!
Thanks.