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 neglected to mention that there were eight $\heartsuit$ among the 32 cards in the deck.
Now the probability of drawing at least one $\heartsuit$ can be calculated by complements $1-P(H=0)$, directly, or by using the Principle of Inclusion and Exclusion.
$$\begin{align}P(H{=}0) & = \dfrac{\dbinom{24}{5}}{\dbinom{32}{4}} \\[2ex] P(H{\geq} 1) & = \dfrac{\dbinom{8}{1}\dbinom{24}{4}+\dbinom{8}{2}\dbinom{24}{3}+\dbinom{8}{3}\dbinom{24}{2}+\dbinom{8}{4}\dbinom{24}{1}+\dbinom{8}{5}}{\dbinom{32}{4}}\\[2ex] P(H{\geq} 1) & = \dfrac{\dbinom{8}{1}\dbinom{31}{4}-\dbinom{8}{2}\dbinom{30}{3}+\dbinom{8}{3}\dbinom{29}{2}-\dbinom{8}{4}\dbinom{28}{1}+\dbinom{8}{5}}{\dbinom{32}{4}}\end{align}$$
You attempted the latter, but only used the first term, which over counts cases where more hearts are drawn.
