Courses   <-  Sean Forman   <-  You Are Here Worksheet on Counting Problems

1. The Pennsylvania license plate has three letters and four numbers. How many possible license plates are there in Pennsylvania?

2. For the Pennsylvania Lottery Daily Number you select a three digit number (001 is a valid selection). How many possible different tickets are possible.

3. If a winning ticket, pays out $500 is this a sucker bet? And if so why is our state government sponsoring this game?

4. If vanity plates can have either a letter, number or a blank in any of the seven positions, how many vanity plates are possible?

5. Next summer for vacation, I'm going to visit five members of the European Union. Ignoring that I may have to cross some countries by EuroRail, if there are 15 members of the European Union, how many routes may I take on my trip? Note that visiting Germany and then France is different from visiting France and then Germany.

6. Before my trip I need to write the embassies for tourist information. How many combinations of five countries may I consider for my trip? Note in this case, I don't care in what order I am visiting France and Germany.

7. If there are 16 members of student government, how many ways can they choose a president, vice president, treasurer and secretary? (problems 5-10 are all related)

8. If the student government has four members from each class, and only seniors can be president, how many ways can they choose the same four offices?

9. The student government has a five-member committee to plan alumni events. How many ways can they choose a committee of five people. Note all committee members are on an equal footing.

10. What if the committees have a single designated leader?

11. What if that designated leader must be a senior?

12. Tough Question: In addition to the alumni committee there is an intramural committee of four members, a cultural affairs committee of four members and an academic honesty committee of three members, with the stipulation that nobody can serve on more than one committee. How many ways can we divide up the student government members into the committees, without a committee leader?

13. For the card problems below, it is often helpful to think about choosing the various aspects of the hand. There are four suits ( $\clubsuit - \spadesuit - \heartsuit - \diamondsuit$ and thirteen ranks $2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - J - Q - K - A$).
    
    How many ways can you select a hand of five cards from a deck of 52?

14. In how many ways can you draw a flush (five cards all of the same suit, choose the suit and then five cards from that suit)?

    $( \clubsuit A - \clubsuit K - \clubsuit Q - \clubsuit 3 - \clubsuit 10)$ (note that these cards are just examples of one possible selection, you will need to find all possible such combinations)

15. In how many ways can you draw a four-of-a-kind (four cards of one rank, and any other card)?

    $( \clubsuit 9 - \heartsuit 9 - \spadesuit 9 - \diamondsuit 9 - other)$

16. In how many ways can you draw a full house (three of one rank and two more of another rank)?

    $( \clubsuit 5 - \heartsuit 5 - \spadesuit 5 - \diamondsuit 10 - \heartsuit 10)$

17. In how many ways can you draw two pair?

    $( \clubsuit 3 - \heartsuit 3 - \spadesuit 8 - \diamondsuit 8 - other)$

18. In how many ways can you draw a three-of-a-kind? Note that the two others can not match in rank, or you would have a full house.

    $( \clubsuit 6 - \heartsuit 6 - \spadesuit 6 - other - other)$