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We are tasked with calculating the probability of getting either a "three of a kind," "four of a kind," or a "royal flush" when dealt a 5-card hand from a standard 52-card deck.

### Data from the image:
- **Royal flush**: 4 outcomes
- **Four of a kind**: 624 outcomes
- **Three of a kind**: 54,912 outcomes
- **Total possible outcomes** (5-card hand): 2,598,960

### Probability Formula:
The probability of an event occurring is given by:

\[
P(E) = \frac{\text{Number of favorable outcomes for event E}}{\text{Total possible outcomes}}
\]

### Calculation:

To find the total probability of getting either a "three of a kind," "four of a kind," or a "royal flush," we sum the favorable outcomes of these events:

\[
\text{Total favorable outcomes} = 4 + 624 + 54,912 = 55,540
\]

Now, we calculate the probability:

\[
P = \frac{55,540}{2,598,960} \approx 0.021374
\]

### Final Answer:
The probability of getting a three of a kind, four of a kind, or royal flush is approximately **0.021374** (rounded to six decimal places).

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Do you need further details or have any questions?

Here are 5 related questions to explore:
1. What is the probability of getting a straight flush or a full house?
2. How does the probability of getting a "two pairs" compare to a "three of a kind"?
3. What is the probability of getting any hand that is not a "no pair"?
4. How does the total probability of a flush compare to a straight?
5. What is the probability of getting no pair at all in a 5-card hand?

**Tip**: When calculating probabilities in card games, always consider whether the deck is shuffled after each hand or whether cards are drawn without replacement!

If you are dealt a 5-card hand (without replacement) from a standard 52-card deck, find the probability of getting three of a kind or four of a kind or a royal flush.

Learn how to calculate the probability of getting three of a kind, four of a kind, or a royal flush when dealt a 5-card hand from a standard deck. This probability calculation uses combinatorics and probability theory to assess favorable outcomes from a deck of 52 cards.

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