To determine the probability of being dealt 3 tens and 2 kings from a deck of 52 cards, we need to follow a step-by-step approach:
1. Calculate the total number of ways to choose any 5 cards from a deck of 52:
- This is calculated using combinations, denoted as [tex]\(\binom{52}{5}\)[/tex].
- [tex]\(\binom{52}{5} = 2,598,960\)[/tex]
2. Calculate the number of ways to choose 3 tens from the 4 tens in the deck:
- This is calculated using [tex]\(\binom{4}{3}\)[/tex].
- [tex]\(\binom{4}{3} = 4\)[/tex]
3. Calculate the number of ways to choose 2 kings from the 4 kings in the deck:
- This is calculated using [tex]\(\binom{4}{2}\)[/tex].
- [tex]\(\binom{4}{2} = 6\)[/tex]
4. Calculate the number of successful outcomes (choosing 3 tens and 2 kings):
- As these are independent events, we multiply the number of ways to choose 3 tens by the number of ways to choose 2 kings.
- Successful outcomes = [tex]\(\binom{4}{3} \times \binom{4}{2} = 4 \times 6 = 24\)[/tex]
5. Calculate the probability of being dealt 3 tens and 2 kings from a deck of 52 cards:
- Probability = (Number of successful outcomes) / (Total number of ways to choose 5 cards)
- Probability = [tex]\(\frac{24}{2,598,960}\)[/tex]
6. Simplify the probability:
- Simplifying [tex]\(\frac{24}{2,598,960}\)[/tex], we get [tex]\(\frac{1}{108,290}\)[/tex].
Therefore, the best answer for the probability of being dealt 3 tens and 2 kings from a deck of cards is:
C. [tex]\(\frac{1}{108,290}\)[/tex]
