To compute the probability of getting a five-card poker hand with three 7's and two 5's, we first need to determine the total number of five-card hands and then the number of hands that meet the specified criteria.
Total number of five-card hands:
In a standard deck of 52 cards, there are \( \binom{52}{5} \) ways to choose five cards.
Number of hands with three 7's and two 5's:
There are \( \binom{4}{3} \) ways to choose three 7's from the four 7's in the deck, and \( \binom{4}{2} \) ways to choose two 5's from the four 5's in the deck. Additionally, there are \( \binom{44}{0} \) ways to choose the remaining cards (since they cannot be 7's or 5's).
So, the total number of hands with three 7's and two 5's is:
\[ \binom{4}{3} \times \binom{4}{2} \times \binom{44}{0} \]
Now, we can calculate the probability:
\[ P(\text{Three 7's and Two 5's}) = \frac{\text{Number of hands with three 7's and two 5's}}{\text{Total number of five-card hands}} \]
\[ P(\text{Three 7's and Two 5's}) = \frac{\binom{4}{3} \times \binom{4}{2} \times \binom{44}{0}}{\binom{52}{5}} \]
Let's calculate it.
