1. Suppose I have 4 shirts, 5 pairs of pants, and 3 pairs of shoes. How many different ways can I select an outfit?
2. Now suppose that I also have 6 ties, and that I may or may not wear a tie. How many different outfits can I select?
3. I work at a diner with a limited menu. We offer 4 appetizers, 6 main courses, 3 desserts, and 4 beverages. Our advertising claims that you can eat a full lunch (appetizer, main course, beverage, dessert) here every day for six months without repeating a meal. Is this claim valid?
4. A math test consists of 25 questions. If you must answer at least 20 of the 25, how many different ways can you take the test, i.e. how many combinations of questions are there? Leave your answer in symbolic form, using sigma notation.
5. Repeat problem 4, but now for the case in which you must answer exactly 20 questions. Find the numeric answer.
For all of the remaining problems, leave your answer in symbolic form unless otherwise instructed.
6. You are in charge of seating arrangements for a Starfleet diplomatic conference. There are 5 Human delegates, 4 Klingons, 3 Romulans, and 3 Cardassians, for a total of 15 delegates. The delegates will be seated around a circular table. Seating rearrangements which move everybody n chairs in the same direction are considered identical.
The Protocol Officer has informed you of the following conditions laid down by the various delegations:
1. No Klingon will sit next to a Romulan.
2. No Cardassian will sit by himself.
3. No Romulan will sit between two Humans.
(a) If all of the conditions are disregarded, how many seating arrangements
are there?
(b) If only the condition of the Cardassians is honored, how many seating
arrangements are there?
(c) If all of the Romulans are seated together (so none is between
two Humans), but other conditions are ignored, how many seating arrangements
are there?
(d) If each of the delegations is seated as a group, how many seating
arrangements are there which meet all the conditions? Find the numeric
answer.
For the following problems, define a 5-string as a sequence of 5 lower-case letters from the English alphabet. (This is a non-standard terminology and is not necessarily used by any other sentient being in the universe.)
7. How many 5-strings are possible?
8. How many 5-strings begin with the letter 'j'?
9. How many 5-strings contain the letter 'q' exactly once?
10. How many 5-strings contain no 'w' in them? How many contain at
least 1 'w'?
11. How many 5-strings can be made if repetition of letters is not
allowed? From this, deduce how many strings must have at least one letter
repeated in them.
12. How many 5-strings contain the letter 'a,' if repetition is not
allowed?
13. How many 5-strings contain the letters "bad" together, in that
order, if repetition is not allowed? Find the numeric answer.
14. How many 5-strings contain the letters "xyz" together, in any order,
if repetition is not allowed? Find the numeric answer.
A standard 52-card deck contains 4 suits, each of 13 cards. A hand consists of a series of cards dealt from a shuffled deck. The order the cards are dealt does not affect the value of a hand.
15. How many 5-card hands are there?
16. How many 5-card hands are all the same suit?
17. How many 5-card hands contain cards from exactly two suits?
18. How many 5-card hands do not contain a 7?
19. How many 5-card hands contain exactly two 7's?
20. How many 5-card hands contain a pair, that is, exactly two cards
of the same denomination? How many contain no two cards of the same
denomination?
21. How many 5-card hands contain a pair of one denomination and a
pair of another denomination?
22. How many 5-card hands contain three of one denomination and a pair
of another denomination?
23. How many 5-card hands contain no cards higher than a 10? (Count
the ace as a low card, i.e. denomination of 1.)
24. How many 13-card hands contain no cards higher than a 10, counting
the ace as a high card?
25. How many 13-card hands contain all four aces?
26. How many 13-card hands contain the 5 highest cards of any suit?
27. How many 13-card hands contain the 5 highest cards of any suit,
and no other cards of that suit?
28. How many 13-card hands contain 8 or more cards of the same suit?
This page Copyleft 1999 Brian K. Hare