If you are dealing from a standard deck of 52 cards, how many different 5 card hands could you have at least one spade
how many have no spades? 39C5
so subtract that from 52C5 possible hands
To determine the number of different 5-card hands that have at least one spade, we can approach the problem by calculating the total number of possible 5-card hands and subtracting the number of hands that have no spades.
Step 1: Calculate the total number of 5-card hands
The total number of 5-card hands that can be formed from a standard deck of 52 cards is given by the combination formula: C(52, 5) = 52! / (5! * (52 - 5)!) = 2,598,960.
Step 2: Calculate the number of 5-card hands with no spades
There are 39 non-spade cards in the deck (13 hearts, 13 diamonds, and 13 clubs). To find the number of 5-card hands with no spades, we can calculate C(39, 5) = 39! / (5! * (39 - 5)!) = 575,757.
Step 3: Calculate the number of 5-card hands with at least one spade
The number of 5-card hands with at least one spade is the difference between the total number of hands and the number of hands with no spades:
Number of hands with at least one spade = Total number of hands - Number of hands with no spades
= 2,598,960 - 575,757
= 2,023,203.
Therefore, there are 2,023,203 different 5-card hands that have at least one spade.
To find out how many different 5-card hands could have at least one spade, we can use the concept of complementary counting.
First, let's find the total number of 5-card hands that can be formed from a standard deck of 52 cards. This can be calculated using the combination formula, denoted as C(n, r), where n is the total number of items and r is the number of items selected at a time. In this case, n = 52 and r = 5.
The formula for combination is given by:
C(n, r) = n! / (r! * (n - r)!)
Using the formula, we can calculate the total number of 5-card hands as:
C(52, 5) = 52! / (5! * (52 - 5)!)
Next, let's calculate the number of 5-card hands with no spades.
There are 39 cards in the deck that are not spades (clubs, diamonds, and hearts). We need to choose 5 cards from these 39. Applying the combination formula again:
C(39, 5) = 39! / (5! * (39 - 5)!)
Now, to find the number of 5-card hands with at least one spade, we can subtract the number of hands with no spades from the total number of 5-card hands:
Number of 5-card hands with at least one spade = Total number of 5-card hands - Number of 5-card hands with no spades
= C(52, 5) - C(39, 5)
And that gives us the answer to your question.