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Counting Problems

1. Twenty people are at a party.

a) How many different pairs of people can shake hands?
b) If 10 partygoers are men and 10 are women, how many distinct handshakes are there if men shake hands only with men and women shake hands only with women?

2. Of five letters, there are 2 T's, 2R's, and 1K.  How many distinct "words" can be formed from these five letters.  (A word is any string of five letters).

3. Suppose 10 tasks have to be performed in a day, one after the other.  Of these there are 5 which can be done only in the morning, and 3 which can be done only in the afternoon.  The others can be done at any time.  Apart from this there are no res trictions on the order in which they may be done.  In how many different orders can the tasks be performed?

4. a) Ten boys are to be grouped into five tennis doubles pairs. In how many ways can this be done?
b) The same ten boys are going to the seaside, two each to Asbury Park, Pt. Pleasant, Wildwood, Ocean City, and Cape May.  In How many ways can this be done?

Explain the difference between the two answers in a) and b).

5. Suppose you are to place square tiles in a 2x4 rectangular pattern on a bathroom wall.  How many different patterns can you make if:

a) You have 15 distinct tiles to use?
b) You have 5 boxes of tiles and tiles in different boxes are different colors?  Each box has at least 8 tiles of the same  color.  Tiles of the same color are indistinguishable from one another.

6. In how many different ways can 6 objects be arranged in a circular pattern?

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