I did draw a tree and found out that this can be done in 24 different ways. But is there a quicker formula? There are a total of four different types of cards, as you know. And we are to draw three of them. Again I would like a formula for finding out how many ways these cards can be drawn without drawing huge trees by hand.
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Use the multiplication principle:
If experiment one can result in $n$ outcomes and if for each outcome of experiment one, experiment two can result in $m$ outcomes, then the total number of outcomes obtained from performing experiment one and then performing experiment two is $nm$. (So this counts ordered outcomes of the form (result from exp. one, result from experiment two).)
This extends to the case of three or more experiments.
So, let's take your card example: draw 3 cards without replacement from 4 distinct cards.
Exp 1: draw the first card
Exp 2: draw the second
Exp 3 draw the third.
Experiment one has 4 outcomes. After drawing the first card, we note that experiment two has 3 outcomes. After drawing the second card, we note experiment three has 2 outcomes.
So the total number of ways to draw three cards is $4\cdot3\cdot2=24$.