Need help on this one,
Let A be a finite set with cardinality |A| = n. Prove that A has exactly
C(n,k) subsets ofcardinality k. ( n and k are arbitrary integers such that
0 =< k = < n)
Hi baz,
If we take k elements from n in all possible arrangements of the k elements,
there will be
many arrangements.
as there are k factors in the above expression,
since by arranging them,
there are n choices for the 1st element,
having chosen one there are (n-1) remaining choices for the 2nd,
(n-2) remaining choices for the 3rd....etc
Since, we are not multiplying by
we can write that as
This has counted all possible groups of k distinct elements from the n
in all possible orders
hence we need to unarrange them, to find the number of subsets
with k elements.
Therefore we divide by k!
is written or or
The number of ways to choose k elements from a total of n is