# Cards on a $4\times 13$ array [closed]

One lays down a deck of $52$ cards face up on a $4 \times 13$ array .One tries to select $13$ cards one from each column with different denominations (not necessarily of different suites)

Find the probability that the selection is possible. Justify your answer I am getting $\frac{4^{13}}{\binom{52}{13}}$.

• It's hard to help if you don't explain how you got your answer. Mar 7, 2018 at 16:26
• When you said 52 cards in 4 x 13 array, Are you arranging it in certain way or does this choice belongs to us????? Mar 7, 2018 at 16:29
• I have 4 possible choices for every 13 denominations which is the numerator Mar 7, 2018 at 16:30
• The choice is random, at least from what I read. Mar 7, 2018 at 16:30
• The choice belongs to the person dealing the cards Mar 7, 2018 at 16:30

Given a set $S$ of $r$ ranks, let $s$ be the number of columns that contain a card of at least one of the ranks in $S$. We cannot have $s < r$ for there are $4r$ cards of the given ranks, and only $4s < 4r$ cards in the columns of $S$. By Hall's Marriage Theorem, there exists a complete matching, which proves the theorem.