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Rosen-Pigeonhole Principle

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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?

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A standard deck of card consists of 52 cards ie clubs, diamonds, hearts and spades 13 of each type.

so 13 X 4 = 52 total cards.

Now the pigeonhole principle states that if n items are put into m containers, with n>m, then at least one container must contain more than one item.

Now let us select diamond first.

we take all cards of diamonds so total 13 cards.

now we take one more card which may any of the remaining 3 to be sure that there are two kinds of cards . till now total 14 cards 14 =  13 diamonds and 1 other(they may be any of clubs,spades or hearts but not diamonds).

now take one more card .

15 = 13 diamonds and 2 other (they may be any of clubs,spades or hearts but not diamonds)

now take one more card

16 = 13 diamonds and 3 others(they may be any of clubs,spades or hearts but not diamonds)

and lastly add one more card

17 = 13 diamonds and 4 other(they may be any of clubs,spades or hearts but not diamonds and there are at least two or more card of type clubs or spades or heart it may be //2 hearts, 1 spades and 1 clubs or any configuration in similar manner like //2spades 1 heart and 1 club // 2 club 1 spades and 1 heart)

so total is 17

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