Question

Cards and probability...
Find the number of possible 5-card hands that contain 4 queens and 1 king.

Answer 1

Consider a 4-card hand made up entirely of queens. How many such hands are there? (Hint: There are only 4 queens in a standard deck.)

What happens when you add one card to the hand? How many possibilities does a 5-card hand have? How many of these extra cards are kings?

Answer 2

It would be 4 choose 4 which is just 1.. Then what would the other part be? @sithsandgiggles

Answer 3

Right, so if you've drawn the 4 queens, that means there are 48 cards remaining in the deck.

4 of these 48 are kings, which means the 5th card has \(\dbinom41=4\) possibilities.

So how many hands contain 4 queens and 1 king?

Answer 4

Oh so it would be 5? @sithsandgiggles

Answer 5

No, you have to use what's called the multiplication rule. Here's a graphical explanation, starting with the knowledge that only one 4-card hand contains all 4 queens:

Each line drawn between the boxes are possibilities that you would count. This means there are 4 hands that contain 4 Q's and 1 K.