Question

How many ways are there to select a 5-card hand from a regular deck such that the hand contains at least one card from each suit. Recall that a regular deck has 4 suits, and there are 13 cards of each suit for a total of 52 cards. Hint: Use Inclusion-Exclusion principle.

Answer 1

There are 685464 ways of selecting the 5-card hand

Explanation:

Since the hand has 5 cards and there should be at least 1 card for each suit, then there should be 3 suits that appear once in the hand, and one suit that apperas twice.

In order to create a possible hand, first we select the suit that will appear twice. There are 4 possibilities for this. For that suit, we select the 2 cards that appear with the respective suit. Since there are 13 cards for each suit, then we have
`{13 \choose 2} = 78` possibilities. Then we pick one card of all remaining 3 suits. We have 13 ways to pick a card in each case.

This gives us a total of 4*78*13³ = 685464 possibilities to select the hand.