To find the probability of being dealt three tens and two jacks from a standard deck of 52 cards, we can break down the problem into two parts: calculating the number of favorable outcomes and the total possible outcomes.

Step 1: Total Possible Outcomes

The total number of ways to choose 5 cards from a deck of 52 cards is given by the combination formula:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Thus, the total possible outcomes are:

$\binom{52}{5}$

Calculating this:

$\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5! \cdot 47!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960$

Step 2: Favorable Outcomes

Next, we need to calculate the number of ways to choose 3 tens and 2 jacks.

• Choosing 3 tens: There are 4 tens in the deck, and we need to choose 3 of them:

$\binom{4}{3} = 4$

• Choosing 2 jacks: There are 4 jacks in the deck, and we need to choose 2 of them:

$\binom{4}{2} = 6$

Step 3: Total Favorable Outcomes

Now, we multiply the number of ways to choose the tens and jacks:

$\text{Total favorable outcomes} = \binom{4}{3} \times \binom{4}{2} = 4 \times 6 = 24$

Step 4: Probability

Finally, the probability of being dealt 3 tens and 2 jacks is:

$P(\text{3 tens and 2 jacks}) = \frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}} = \frac{24}{2,598,960}$

Step 5: Simplifying the Fraction

Now we simplify the fraction:

$\frac{24}{2,598,960} = \frac{1}{108,740}$

Thus, the probability of getting three tens and two jacks is:

$\boxed{\frac{1}{108740}}$

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How would the probability change if you wanted four tens and one jack?
2. What is the probability of getting a flush in the same hand?
3. How many different combinations of five cards can you form from the deck?
4. How does the probability differ when drawing from a deck with jokers included?
5. What is the probability of getting a full house (three of one kind and two of another)?

Tip: When calculating probabilities, always consider both the favorable outcomes and the total outcomes carefully!