How many ways are there to select a 5-card hand from a regular deck such that the hand contains at least one card from each suit
1 answer:
Answer:
There are 685464 ways of selecting the 5-card hand
Step-by-step explanation:
Since the hand has 5 cards and there should be at least 1 card for each suit, then there should be 3 suits that appear once in the hand, and one suit that apperas twice.
In order to create a possible hand, first we select the suit that will appear twice. There are 4 possibilities for this. For that suit, we select the 2 cards that appear with the respective suit. Since there are 13 cards for each suit, then we have possibilities. Then we pick one card of all remaining 3 suits. We have 13 ways to pick a card in each case.
This gives us a total of 4*78*13³ = 685464 possibilities to select the hand.
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Answer:
Step-by-step explanation:
Let:
This is and exact equation, because:
So, define f(x,y) such that:
The solution will be given by:
Where C1 is an arbitrary constant
Integrate with respect to x in order to find f(x,y):
Where g(y) is an arbitrary function of y.
Differentiate f(x,y) with respect to y in order to find g(y):
Substitute into
Integrate with respect to y:
Substitute g(y) into f(x,y):
The solution is f(x,y)=C1
Solving y using quadratic formula:
First, let's find g(4). After we find it, we can substitute the value into .
To find g(4), we need to substitute x = 4 into the function , as shown below:
Now that we have found g(4) = 9, we can substitute this value in for in
.
f(g(4)) is equal to 28.