Count the number of elements in the event

Question

A state lottery game consists of choosing one card from each of the four suits in a standard deck of playing cards. (There are 13 cards in each suit)

Count the number of elements in the event than an ace, a king, a queen, and jack are chosen.

Answer 1

Let's call the valued cards royal. In a pack of 52 cards there will be 16 royals. The odds of picking one of them is 16 in 52. That leaves 51 cards containing 15 royals. So the odds of picking another royal is 15 in 51. So we continue until only 37 remain, one of which is the remaing royal. The odds of picking all 16 royals in any order is therefore 16/52*15/51*...1/37 to 1 against. The number of events is the reciprocal of this which is 52C16 in nCr format. This evaluates to 10,363,194,502,115. Go for it - you can't lose!! Not!