The 4 card hand with a fixed ace

Hi. You have misunderstood the combinatorial task needed here :)

1) So, you have 4-card hand out of 52 cards and want one of the cards to be an ace. All possible outcomes for 4-card hand out of 52 cards are

52C4 = 52*51*50*49/4! = 270 725.

If you think about all the possible combinations excluding any ace (all aces are 4, i.e. our deck decreases to 48) in the 4-card hand - they are 48C4 = 48*47*46*45/4! = 194 580. So, the answer you are looking for is:

{All possible combinations of 4-card hand out of 52} minus

{All possible combinations of 4-card hand out of 48} - we have "removed" all 4 aces i.e. 270 725 - 194 580 = 76 145 which is the answer you mentioned.

This valid as you may see for any denomination of a card deck - aces, jacks, 7s and etc...

2) Give me please, more clear and concise mathematical definition of the problem conditions(esp. def of a "word") - which letters to repeat or not any of them to be repeated, so that I will be able to help.. I think about permutations with repeat, if you know what I mean.