Compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings. (Give your answer as a fraction.)
4/52 * 3/51 * 2/50 * 4/49 * 3/48 = ?
2995/10000
To compute the probability of drawing three Aces and two Kings from a standard deck of playing cards, we need to understand the total number of possible outcomes and the number of favorable outcomes.
Total Number of Outcomes:
When drawing five cards from a deck of 52 playing cards, there are a total of 52C5 possible outcomes. "52C5" represents the number of ways to select 5 cards from a deck of 52, without regard to their order. Mathematically, 52C5 can be computed as:
52C5 = (52! / (5!(52-5)!))
Number of Favorable Outcomes:
To select 3 Aces and 2 Kings, we need to consider the number of favorable outcomes. There are 4 Aces in the deck and 4 Kings, so the number of ways to select 3 Aces from 4 and 2 Kings from 4 is:
4C3 * 4C2 = (4! / (3!(4-3)!)) * (4! / (2!(4-2)!)) = 4 * 6 = 24
Probability:
The probability of obtaining 3 Aces and 2 Kings is the ratio of favorable outcomes to total outcomes:
Probability = Number of Favorable Outcomes / Total Number of Outcomes
Probability = 24 / (52C5)
Now let's calculate the values:
First, we need to evaluate 52C5:
52C5 = (52! / (5!(52-5)!))
= (52! / (5! * 47!))
= (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1)
= 2,598,960
Now, substitute the values into the probability calculation:
Probability = 24 / 2,598,960
Therefore, the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings is 24/2,598,960, which simplifies to 1/108,290.