From a standard 52 card deck, how many 5-card hands will have all hearts?
There are only 13 hearts, so there are
C(13,5) = 13!/5!8! = 1287 ways to pick them
To find the number of 5-card hands that will have all hearts from a standard 52-card deck, we need to determine the number of ways to choose 5 cards from the 13 hearts in the deck.
The number of ways to choose "k" items out of "n" items is given by the binomial coefficient, denoted as "n choose k" or (nCk).
In this case, we need to calculate the number of ways to choose 5 cards out of the 13 hearts, so we have (13C5).
The formula for the binomial coefficient is:
(nCk) = n! / (k! * (n-k)!)
Where "!"" denotes the factorial of a number, which means the product of an integer and all the positive integers below it.
Plugging in the values for our problem, we have:
(13C5) = 13! / (5! * (13-5)!)
Simplifying:
(13C5) = 13! / (5! * 8!)
Now, we need to evaluate these factorials:
13! = 13 * 12 * 11 * 10 * 9 * 8!
5! = 5 * 4 * 3 * 2 * 1
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Substituting these values into the equation:
(13C5) = (13 * 12 * 11 * 10 * 9 * 8!) / (5 * 4 * 3 * 2 * 1 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Simplifying further:
(13C5) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)
Evaluating this expression, we find that:
(13C5) = 1287
Therefore, there are 1287 different 5-card hands that will have all hearts from a standard 52-card deck.