2 We are creating a new card game with a new deck Unlike the

2) We are creating a new card game with a new deck. Unlike the normal deck that has 13 ranks (Ace through King) and 4 Suits (hearts, diamonds, spades, and clubs), our deck will be made up of the following.

Each card will have:
i) One rank from 1 to 12.
ii) One of 9 different suits.

Hence, there are 108 cards in the deck with 12 ranks for each of the 9 different suits, and none of the cards will be face cards! So, a card rank 11 would just have an 11 on it. Hence, there is no discussion of \"royal\" anything since there won\'t be any cards that are \"royalty\" like King or Queen, and no face cards!

The game is played by dealing each player 5 cards from the deck. Our goal is to determine which hands would beat other hands using probability. Obviously the hands that are harder to get (i.e. are more rare) should beat hands that are easier to get.

How many different ways are there to get exactly 1 pair (i.e. 2 cards with the same rank)?

Solution

From above explanation :

There are 12 ranks.

There are 9 different suits.

We need to pick a pair of cards of same rank

The first card can be of any rank. The next card must be of same rank. second card having same number is :

Since there are 8 more suites, there can be 8 more cards of same number. Total cards after already picking first card is 108.

So number of ways of picking a pair of cards with same rank = 108*8 = 864