Compute number of combinations

Question

I got really stuck with this task:

Nine of ten cards, among which there is an ace of hearts, are distributed to three players so that the first one receives 3, the second - 4, and the third - 2 cards. How many cards combinations exist, where an ace of hearts gets to a third player?

I think that number of combinations formula is $c_{9}^{1} * c_{3}^{1} * c_{2}^{1} = 6$

Am I right? I will be so grateful for your help!)

Answer 1

Just give the ace of hearts to the third player first. Then allocate the rest of the hands:

• The third player gets a second card to complete their hand in $\binom91=9$ ways, leaving $8$ cards
• The second player gets their hand in $\binom84=70$ ways, leaving $4$ cards
• The first player gets their hand in $\binom43=4$ ways

Thus there are $9\times70\times4=2520$ ways to distribute the cards.