Discrete Math Probability Permutations and Combinations a Ho

Discrete Math: Probability, Permutations and Combinations

a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen?

b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and at least two of the 13 kinds are chosen?

c) How many cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of the same kind?

d) How many cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?

Solution

Solution:

a) we suppose that in the worst case, we draw 48 cards other than aces. Then the next two cards are aces.

Number of cards = 48+2=50

c) there are four kind of cards.(spades,heart,diamond,clubs)

We select 4 cards of each kind. And 1 card of any kind, so there will be atleast 2 cards of same kind.

So there must be 5 pick up to ensure the same kind condition.

Answer: 4+1=5

d)We can only have 13 cards of one kind. So, in the worst case we pick these 13 cards all of same kind.

Now, if we pick the next card of a different kind, we can do like this only 3 more times- as there are only 4 different kinds. The fourth time we do, we must repeat a kind. So, 13+4 = 17 is the answer.