A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and c

Question

A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each containing 13 different values (Ace, 2, 3, …, 10, J, Q, K). If you draw some number of cards at random you might or might not have a pair (two cards with the same value) or three cards all of the same suit. However, if you draw enough cards, you will be guaranteed to have these. For each of the following, find the smallest number of cards you would need to draw to be guaranteed having the specified cards. Prove your answers.

A) Three of a kind (for example, three 7's).

B) A flush of five cards (for example, five hearts).

C) Three cards that are either all the same suit or all different suits.

Explanation / Answer

In a standard deck of 52 cards, we have 13 different types of card in 4 different suits.

Pigeonhole Principle: If n items are put into m containers, where n>m, then atleast one of the container will have more than one item.

A) Suppose we choose an entire suit of 13 different cards. We add another 13 cards of a different suit without forming a group of 'three of a kind'. Now, on drawing any of the remaining card will give a group of three cards of same kind. Therefore, we must draw atleast 27 cards to be sure to obtain this result.

B) In an extreme case, one may draw 4 cards of each suit not giving a flush of 5 cards. But an additional draw will give the desired result. So, to be sure to obtain a flush of five cards, one must draw 4+4+4+4+1=17 cards.

C) Lets say 2 cards of one suit and 2 of another is drawn, one more selection of card will give a group of 3 cards of same suit or of all different suits. Thus, 5 cards must be drawn here.

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