Only one correct solution was received.  Congratulations to Lou Cairoli.

The result follows easily as an application of Phillip Hall's so-called "Marriage Theorem".  However, in this case, it is also easy to give an algorithm which selects four coins --- one from each row, one of each denomination --- from the array of coins: 

If all the coins in one of the rows are of the same denomination, say all pennies or all nickels, then select one of those coins from that row and remove all the other coins of that type from the array.  In no such row exists, select any coin in the array and remove all the other coins of that denomination from the array.  Repeat these steps until the array is empty.

Note that there are 63,063,000 possible distinct arrangements of coins.  If you had checked one per second, without pause, 24 hours a day, 7 days a week, it would take almost two years to check them all!  

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©2005 Alberto L. Delgado