Question

You are dealt five cards from a standard deck of 52 playing cards (A full house consists of three of one kind and two of another. For example, A A A 5-5 and K-K-K 10-10 are full houses) (a) in how many ways can you get a full house? ______ Ways (b) in how many ways can you get a five card combination containing two jacks and three aces ___ ways

Answer 1

There are 3744 ways to get a full house and 24 ways to get a five-card combination containing two jacks and three aces in a standard deck of 52 cards.

Step-by-step explanation:

To calculate the number of ways to get a full house in a standard deck of 52 playing cards, consider the two types of cards you need: a triplet and a pair. First, select the rank of the triplet (13 ways) and then choose three suits from the four available for this rank (4 choose 3 ways). Next, select the rank of the pair (12 remaining ways, since one rank is already taken by the triplet) and then choose two suits from the four available for this rank (4 choose 2 ways). The calculation is as follows:

• 13 (ranks for the triplet)
  
• 4 choose 3 (suits for the triplet)
  
• 12 (remaining ranks for the pair)
  
• 4 choose 2 (suits for the pair)
  

Therefore, the number of ways to get a full house is 13 × (4 choose 3) × 12 × (4 choose 2) which equals 13 × 4 × 12 × 6 = 3744 ways.

To calculate the number of ways to get a five-card combination containing two jacks and three aces, select two suits for the jacks (4 choose 2 ways) and three suits for the aces (4 choose 3 ways). The calculation is:

• 4 choose 2 (suits for the jacks)
  
• 4 choose 3 (suits for the aces)
  

The number of ways to achieve this specific combination is (4 choose 2) × (4 choose 3) which equals 6 × 4 = 24 ways.