Try It Now

Exercises

1. In horse racing, to bet the "daily double" is to select the winners of the first two races of the day. You win only if both selections are correct. In terms of the number of horses that are entered in the first two races, how many different daily double bets could be made?
   
   
3. A certain shirt comes in four sizes and six colors. One also has the choice of a dragon, an alligator, or no emblem on the pocket. How many different shirts could you order?
   
   
5. The Pi Mu Epsilon mathematics honorary society of Outstanding University wishes to have a picture taken of its six officers. There will be two rows of three people. How many different ways can the six officers be arranged?
   
   
7. A clothing manufacturer has put out a mix-and-match collection consisting of two blouses, two pairs of pants, a skirt, and a blazer. How many outfits can you make? Did you consider that the blazer is optional? How many outfits can you make if the manufacturer adds a sweater to the collection?
   
   
9. 1. Suppose each single character stored in a computer uses eight bits. Then each character is represented by a different sequence of eight 0's and 1's called a bit pattern. How many different bit patterns are there? (That is, how many different characters could be represented?)
   2. How many bit patterns are palindromes (the same backwards as forwards)?
   3. How many different bit patterns have an even number of 1's?
      
      
11. 1. Let A = {1, 2, 3, 4} . Determine the number of different subsets of A.
    2. Let A = {1 , 2, 3, 4, 5}. Determine the number of proper subsets of A.
       
       
13. Consider three persons, A, B, and C, who are to be seated in a row of three chairs. Suppose A and B are identical twins. How many seating arrangements of these persons can there be:
    
    1. If you are a total stranger?
    2. If you are A and B's mother?
       
       This problem is designed to show you that different people can have different correct answers to the same problem.
       
       
15. Suppose you have a choice of fish, lamb, or beef for a main course, a choice of peas or carrots for a vegetable, and a choice of pie, cake, or ice cream for dessert. If you must order one item from each category, how many different dinners are possible?
    
    
17. A questionnaire contains six questions each having yes-no answers. For each yes response, there is a follow-up question with four possible responses.
    
    1. Draw a tree diagram that illustrates how many ways a single question in the questionnaire can be answered.
    2. How many ways can the questionnaire be answered?
       
       
19. How many ways can you separate a set with n elements into two nonempty subsets if the order of the subsets is immaterial? What if the order of the subsets is important?

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Solutions

1. Answer: If there are m horses in race 1 and n horses in race 2 then there are m · n possible daily doubles.
   
   
3. Answer: 72 = 4 · 6 · 3
   
   
5. Answer: 720 = 6 · 5 · 4 · 3 · 2 · 1
   
   
7. Answer: If we always include the blazer in the outfit we would have 6 outfits. If we consider the blazer optional then there would be 12 outfits. When we add a sweater we have the same type of choice. Considering the sweater optional produces 24 outfits.
   
   
9. Answer:
   1. 2⁸ = 256
   2. 2⁴ = 16. Here we are concerned only with the first four bits, since the last four must be the same.
   3. 2⁷ = 128, you have no choice in the last bit.
      
      
11. Answer:
    
    1. 16
    2. 31
       
       
13. Answer:
    
    1. 3
    2. 6
       
       
15. Answer: 18
    
    
17. Answer:
    1. 
    2. 5⁶
       
       
19. Answer: 2^n − 1 − 1 and 2ⁿ − 2