A flush is 5 cards of the same suit.  I don't know if
you want to consider a straight flush as a flush or not.

Choose the suit for the flush 4 ways.
Choose the ranks: 13 cards choose 5. That's 13C5 

Answer: 4*(13C5) = 4(1287) = 5148 

There are 5148 possible flushes, including straight flushes.

Straight flushes can be a set of any of these 10

{A,2,3,4,5}
{2,3,4,5,6}
{3,4,5,6,7}
{4,5,6,7,8}
{5,6,7,8,9}
{6,7,8,9,10}
{7,8,9,10,J}
{8,9,10,J,Q}
{9,10,J,Q,K}
{10,J,Q,K,A}

There are 4 suits that each could be, so
there are 10*4 = 40 straight flushes.

So the answer is 5148 if you are counting straight flushes,
but only 5148-40 = 5108 if you are not 

Edwin