From a standard 52 card deck, how many 5 card hands will have 2 clubs and 3 hearts?
Well, let me use my clown math skills here. So, we need to choose 2 clubs out of 13, and 3 hearts out of the remaining 13 (since we already used 2 clubs).
Now, the number of ways to choose k items out of a set of n items is given by the combination formula, nCk, which is equal to n! / (k!(n-k)!), where "!" represents factorial.
So, in this case, we have 13C2 ways to choose 2 clubs, which is 13! / (2!(13-2)!), and 13C3 ways to choose 3 hearts, which is 13! / (3!(13-3)!).
To find the total number of 5-card hands with 2 clubs and 3 hearts, we multiply these two together: (13! / (2!(13-2)!) ) * (13! / (3!(13-3)!)).
Now, let's see if anyone has a calculator... or we could just round it to the nearest clown number. I say it's somewhere around 420. Why? Because 420 makes everything funnier, right?
To find the number of 5-card hands that will have 2 clubs and 3 hearts, you can break down the problem into smaller steps.
Step 1: Calculate the number of ways to choose 2 clubs from the deck.
Since there are 13 clubs in a standard 52-card deck, you can choose 2 clubs in C(13,2) ways.
C(13,2) = (13!)/(2!(13-2)!) = (13 * 12)/(2 * 1) = 78
Step 2: Calculate the number of ways to choose 3 hearts from the deck.
Since there are 13 hearts in a standard 52-card deck, you can choose 3 hearts in C(13,3) ways.
C(13,3) = (13!)/(3!(13-3)!) = (13 * 12 * 11)/(3 * 2 * 1) = 286
Step 3: Calculate the number of ways to choose the remaining 5 - (2 + 3) = 5 - 5 = 0 cards from the remaining 52 - (13 + 13) = 52 - 26 = 26 cards (the non-club and non-heart cards).
Since there are no remaining cards to choose, there is only 1 way to choose 0 cards.
Step 4: Multiply the results from steps 1, 2, and 3 to find the total number of 5-card hands with 2 clubs and 3 hearts.
Total number of 5-card hands = C(13,2) * C(13,3) * C(26,0) = 78 * 286 * 1 = 22308
Therefore, there will be 22,308 5-card hands with 2 clubs and 3 hearts.
To find the number of 5-card hands that have 2 clubs and 3 hearts from a standard 52-card deck, you can break down the solution into two steps:
Step 1: Calculate the number of ways to choose 2 clubs from the deck.
Since there are 13 clubs in a deck, you can choose 2 clubs out of 13. This can be calculated using the combination formula, denoted as "n choose k" or "nCk," which is given by:
nCk = n! / (k! * (n-k)!)
Using this formula, the number of ways to choose 2 clubs from 13 can be calculated as:
13C2 = 13! / (2! * (13-2)! )
= 13! / (2! * 11! )
= (13 * 12) / (2 * 1)
= 156 / 2
= 78
So, there are 78 ways to choose 2 clubs from a deck of 13 clubs.
Step 2: Calculate the number of ways to choose 3 hearts from the remaining deck.
After choosing 2 clubs, you are left with 50 cards in the deck (39 non-heart cards and 11 hearts). Now, you need to choose 3 hearts from these 11 remaining hearts. Using the same combination formula as before, you can calculate this as:
11C3 = 11! / (3! * (11-3)!)
= 11! / (3! * 8!)
= (11 * 10 * 9) / (3 * 2 * 1)
= 990 / 6
= 165
So, there are 165 ways to choose 3 hearts from the remaining deck of 11 hearts.
Finally, you can multiply the results from Step 1 and Step 2 to get the total number of 5-card hands that have 2 clubs and 3 hearts:
Total number of hands = Number of ways to choose 2 clubs * Number of ways to choose 3 hearts
= 78 * 165
= 12,870
Therefore, there are 12,870 5-card hands that have 2 clubs and 3 hearts from a standard 52-card deck.
number of hands with 2 clubs and 3 hearts
= C(13,2) * C(13,3)
= 22308