Probability, Why is this wrong? So we have a deck of 32 cards , let's suppose we take 5 cards at the same time,
then we search for the probability of having at least one ♥ heart card
for this we can do 1- p(having no heart card) which translate to 
1 - (24 C 5)/(32 C 5)
which will give you approximately 0.77
so far it's all good,
the problem start here : 
we can (or can't?) also consider that "having at least one heart" as : "pick one heart then any other 4 cards"
which translate (or not?) to :
((8 C 1)*(31 C 1))/|Ω|
which give 1.25, which is obviously wrong
The question is, where is the mistake is my process?

You overcount the hands with more than one heart in the second method.
If the hand contains, for example, both the ace and king of hearts and three non-hearts, such hands are counted twice, because you may have first selected the heart ace as the heart, and the king among the 4 out of remaining 31. Or the other way round. Should a hand contain more than two hearts the overcounting becomes more severe, but those hands are not very numerous.
A correct variant is the following.
Assume that the hand has $k$ hearts and hence $5-k$ non-hearts. The number of such hands is clearly
$$
{}_8C_k\cdot{}_{24}C_{5-k}=\binom 8 k\cdot\binom{24}{5-k}
$$
depending on how you want to write the binomial coefficients.
Thus the number of hands with at least one heart is
$$
\sum_{k=1}^5{}_8C_k\cdot{}_{24}C_{5-k}=158872.
$$
As a reality check we see that the total number of hands is $\binom{32}5=201376$, and the number of hands with no hearts $\binom{24}5=42504=2013776-158872.$
