There are 15,820,024,220 possible ten-card hands in Gin Rummy
John Scarnes lists the number of hands as 15,820,024 in John Scarne's Encyclopedia of Card Games 1983 ISBN 0-06-273155-6 page 97.
John Scarnes' number, however, is incorrect although it looks very similar to a correct number of 15,820,024,220
Below is the formula calculating a number of 10-card gin rummy hands taken from a 52-card deck. The number of cards is counted as combinations as opposed to permutations so the order of the cards in hand does not matter.
10_Card_Hands = 52! / (10! * 42!)
10_Card_Hands = 15,820,024,220
If you are further interested in the theory of combinations and permutaions, please refer to TheMathPage