20 Principle of Counting a Explain why among a group of 60 f

#20 Principle of Counting

a) Explain why among a group of 60 foreign exchange students from the United States, at least 2 came from the same state.

b) What is the minimum number of cards that must be drawn from a shuffled standard deck of 52 cards to gurantee that there will be at least one pair? Provide an example to show that is one less than this number is drawn, a pair need not come up.

Solution

a. This result follows from the pigeonhole principle. Even if we tried to assign students in the sparsest way (ie, one student per state), one of the 50 states must have more than one student because there are greater than 50 students.

b. In any deck of 52 cards, there are 13 different denominations and 4 suits. We must draw at least 14 cards to ensure that one denomination occurs twice. An example is the 13-card hand comprising an ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, where the cards may be from any suit. (Except if the deck came from the French revolution, in which the latter three would be les libert