To calculate the probability of being dealt a flush in diamonds (not a royal flush or a straight flush) in a 5-card poker hand, we can follow these steps:

Total possible hands in poker:

The total number of possible 5-card hands from a deck of 52 cards is computed using the combination formula $C(n, r)$, which gives the number of ways to choose $r$ objects from $n$ without regard to the order:

$C(52, 5) = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!}$

This gives the total number of possible hands in a 52-card deck.

Number of ways to get a diamonds flush (non-royal, non-straight):

To get a flush in diamonds, we must select 5 cards all from the 13 diamonds. The number of ways to choose 5 cards from the 13 diamonds is:

$C(13, 5) = \frac{13!}{5!(13-5)!} = \frac{13!}{5!8!}$

However, we must exclude any royal flush (Ace, King, Queen, Jack, 10) and any straight flush (five consecutive cards of the same suit). There is only 1 royal flush (Ace, King, Queen, Jack, 10 of diamonds) and 9 possible straight flushes (e.g., Ace to 5, 2 to 6, etc.).

Thus, the number of flushes without a straight or royal flush is:

$C(13, 5) - 10 = \text{valid flushes}$

Probability calculation:

Finally, the probability is:

$P(\text{flush}) = \frac{\text{valid flush hands}}{C(52, 5)}$

Let me calculate the exact value for you.The probability of being dealt a diamonds flush (not a royal flush or a straight flush) in a 5-card poker hand is approximately $0.00049135$ (rounded to eight decimal places).

Do you want further details or have any questions?

Here are some related questions for further exploration:

1. What is the probability of being dealt a royal flush in diamonds?
2. How does the probability of getting a flush change for a different suit?
3. What is the probability of being dealt a straight flush in any suit?
4. How do the odds of getting a flush compare with the odds of getting a full house?
5. What is the probability of getting a four-of-a-kind hand in poker?

Tip: In poker, the odds of specific hands depend heavily on the combinations of cards available, so understanding how to calculate combinations is key!