Math 141, Sem. I, 1998-99 (R.A. Brualdi)
Handout 17 (November 2, 1998)
Topic: Counting problems


1. A ZIP-code consists of 5 digits, such as 90113. How many possible ZIP-codes
are there? 

2. If the population of the USA is taken to 250 million, how many digits would
a ZIP code need to have in order to give each resident her/his own ZIP code? 

3. If a ZIP code consisted of digits and letters, with any symbol available in
each spot, how many spaces would ZIP codes need to have to give everyone
her/his own ZIP code? [Use the 250 million pop. figure.]

4. Code numbers on cases of vegetable soup look like AT-4178, two letters
followed by four digits. How many  code numbers are there?

5. Our class contains 14 students - eleven women and three men.  How many
different "committees" of three people could be formed? How many with only
men?  With only women? With one man and two women? With two men and one
woman? 

6. A deck of cards contains 52 cards, 13 of each of the four suits. A HAND
is a selection of 5 cards from the deck, ignoring order of draw.

a) How many HANDs are there?

b) How many HANDs contain only clubs?

c) How many HANDs contain only cards of the same suit? (Such a HAND is called
a flush.) 

d) How many HANDs contain only face cards (kings, queens and jacks)? 

e) How many HANDs are full houses (three of one kind and two of another)?

f) How many straights (five consecutive cards of any suits - mixed or not) are
there? 

g) How many straight flushes (five consecutive cards of the same suit) are
there? 

7. A supermarket "scratch and win" game card has 16 spots covered with
silver film. You can scratch off 6 spots to see if you have won anything. 
How many ways can you choose your spots (order in which you choose them
doesn't matter)?

8. How many ways are there to have 8 non-attacking rooks on a 8 by 8
chessboard? [rooks can attack horizontally or vertically] Try the roblem
first for 2 by 2, 3 by 3, 4 by 4 boards.

9. Consider a 2 by n "chessboard" (total of 2n squares arranged in two
rows and n columns). For n=2, 3, 4, 5 determine the number of ways to tile
the board with n non-overlapping dominoes (1 by 2 pieces). E.g. one way is
to put all the dominoes vertical. Find a method to compute the number of
ways in general (not asking for a specific formula here) .