# What are the odds of drawing five specific cards from a sixty card deck? [duplicate]

In Yugioh, you start out with a five card hand. You have a 40-60 card deck. I run a 60 card Exodia deck, and twice I have drawn all five pieces of Exodia in my starting hand. For those who don’t play, having all five pieces in your hand at one time causes you to immediately win the game. I would like to know what the actual odds of that happening once, let alone twice.

• Voting to close as duplicate - while the other question includes each draw phase, the first answer deals with the start of game hand draw - though it like most YGO questions uses the optimal deck size of 40 cards. Dec 11, 2020 at 15:22

1 in 5,461,512

This is a nCr question, you have n elements (60) and you are picking r of them (5). Since each piece of Exodia is Limited to 1 copy per deck, and you are looking for 5 specific cards out of a 60 card deck, the math is a straightforward nCr calculation:

60 C 5 = 60! / ((60 - 5)! * 5!) = 5,461,512.

There are 5,461,512 possible combinations of 5 different cards you can make out of 60, since you are looking for 5 unique cards to all be drawn, that means you want 1 combination out of those, thus 1 in 5,461,512.

If this was a deck before all the parts were limited to single copies (or kitchen table, ignoring the deck limit rules) then that would give a different answer.*

Generally in games like this people stick to the minimum deck size, a lot of new players to TCGs like to add more into their decks, but this shows why that's a bad idea. If this was a 40 card Exodia deck instead:

40 C 5 = 40! / ((40 - 5)! * 5!) = 658,008

At the minimum deck size, getting all 5 pieces in the opening hand is over 8 times more likely.

*To figure the difference out with multiple copies of the same cards in the deck, you change the left side, multiplying the number of copies each wanted card, for these 5 restricted cards that gives 1 * 1 * 1 * 1 * 1 = 1. If you could have the normal 3 copies of each piece in your deck that would be 3 * 3 * 3 * 3 * 3 = 243 possible combinations.

In 60 cards that means 243 in 5,461,512 or 1 in a little under 22,500. for 40 cards that is 243 in 658,008 or 1 in a little over 2,200.