Hello Forum, I need help with the following,
Five cards are chosen at random from an ordinary deck of playing cards. In how many ways can the cards be chosen under each of the following conditions?
a. All are hearts. b. All are the same suit.
c. Exactly three are kings. d. Two or more are aces.
Hello, AlgebraicallyChallenged!
Since you seem to be unable to do any of these,
. . I'll give you a walk-through ... with baby steps.
I assume you know about Combinations.
We want 5 of the 13 Hearts.Five cards are chosen at random from an ordinary deck of playing cards.
In how many ways can the cards be chosen under each of the following conditions?
a. All are Hearts.
There are: . ways to get Five Hearts.
b. All are the same suit ( a "flush").
There are 4 choices for the suit.
And there are: . to get 5 cards of that suit.
Therefore, there are: . flushes.
c. Exactly three are kings.
There are 4 Kings ... and 48 non-Kings.
We want 3 of the 4 Kings: . ways.
We want 2 of the 48 non-Kings: . ways.
Therefore, there are: . ways to get Three Kings.
This can be done by the "complement" approach; I'll do it head-on.d. Two or more are aces.
There are 4 Aces and 48 Others.
There are three cases to consider . . .
Two Aces
There are ways to choose the two Aces.
There are ways to get three Others.
. . Hence, there are: . ways to get Two Aces.
Three Aces
There are ways to choose the three Aces.
There are ways to get two Others.
. . Hence, there are: . ways to get Three Aces.
Four Aces
There ae way to choose the four Aces.
There are ways to get one Others.
. . Hence, there are: . ways to get Four Aces.
Therefore, there are: . ways to get two or more Aces.