To determine the probability of being dealt 3 tens and 2 kings from a standard deck of 52 cards, let's go through the steps carefully:
1. Total Number of Ways to Choose 5 Cards Out of 52:
We need to calculate how many ways we can choose any 5 cards from a deck of 52 cards. This is a combination problem, and the formula for combinations is given by:
[tex]\[
\binom{52}{5} = \frac{52!}{5!(52-5)!}
\][/tex]
2. Number of Ways to Choose 3 Tens Out of 4:
There are 4 tens in a deck (ten of hearts, ten of diamonds, ten of clubs, ten of spades). We need to find out the number of ways to choose 3 out of these 4 tens:
[tex]\[
\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4
\][/tex]
3. Number of Ways to Choose 2 Kings Out of 4:
Similarly, there are 4 kings in a deck. We need to find the number of ways to choose 2 out of these 4 kings:
[tex]\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6
\][/tex]
4. Number of Favorable Outcomes:
The number of favorable ways to choose 3 tens and 2 kings simultaneously is the product of the combinations of tens and kings:
[tex]\[
\text{Favorable Ways} = \binom{4}{3} \cdot \binom{4}{2} = 4 \cdot 6 = 24
\][/tex]
5. Probability:
Finally, the probability of being dealt 3 tens and 2 kings is the ratio of favorable outcomes to the total number of possible outcomes:
[tex]\[
\text{Probability} = \frac{\text{Favorable Ways}}{\binom{52}{5}}
\][/tex]
Given the answer from calculations, the probability is approximately:
[tex]\[
9.23446301597562 \times 10^{-6}
\][/tex]
6. Matching Probability to the Given Choices:
The numerical value [tex]\(9.23446301597562 \times 10^{-6}\)[/tex] is equivalent to [tex]\(\frac{1}{108,290}\)[/tex] when expressed as a simplified fraction.
Thus, the best answer for the question is:
[tex]\[
\boxed{\frac{1}{108,290}}
\][/tex]
So, the correct option is:
B. [tex]\(\frac{1}{108,290}\)[/tex]
