Practice Homework (Solutions) -- Combinatorics

  1. A girl has five skirts and eight blouses. How many skirt-blouse outfits does she own? 40

  2. How many permutations are there of the letters a,b,c. List them. P(3,3) = 6.
    1. a,b,c
    2. a,c,b
    3. b,a,c
    4. b,c,a
    5. c,a,b
    6. c,b,a

  3. A theater will show a new movie called Middlemen. If an employee is given the letters (all capital) for the marquis, in how many ways can he misspell the movie title. 9!/(2!*2!*2!) - 1 = 45359

  4. Write the following using factorials:
    1. 12*11*10*9 = 12!/8!
    2. 30*29 = 30!/28!

  5. How many 5-digit numbers are there where the first digit is non-zero? 9*10*10*10*10 = 90000

  6. In how many ways can five science books, three history nooks, and four mathematics books be arranged on a shelf if books of the same subject must remain together? First order the subjects and then order the books within subject: 3! * 5! * 3! * 4!= 103680

  7. Evaluate each of the following:
    1. C(10,3) = 10 * 9 * 8 / (3 * 2 * 1) = 120
    2. C(90,88) = 90 * 89 / 2 * 1 = 4005
    3. C(13,3) * C(12,2) / C(25,5) = 0.3553

  8. How many committees of four can be formed from sixteen people? C(16,4) = 16 * 15 * 14 * 13 / (4 * 3 * 2 * 1) = 1820

  9. Suppose you have a bowl of alphabet soup?
    1. How many combinations of five distinct letters may be picked up on a spoon? C(26,5) = 65,780
    2. Suppose there is one pair of identical letters and three other distinct letters? C(26,1) * C(25,3) = 59800

  10. Steve and 10 of his friends are choosing sides for a basketball game. There are five on each team, and the excluded person is the umpire.
    1. How many different ways are there to divide the friends into teams if Steve is the umpire? C(10,5)/2 = 126
    2. How many different ways are there to divide the friends into teams if Steve is not the umpire? = 10 * 126 = 1260

  11. A flush in poker consists of five cards all of the same suit. How many ways are there to get a flush? C(4,1) * C(13,5) = 5148

  12. A jar contains six black and nine white balls. How many ways are there of selecting four balls if
    1. two are black? C(6,2) * C(9,2) = 540
    2. all four are the same color? = C(6,4) + C(9,4) = 141

  13. A test has twelve questions and you must answer ten. How many choices do you have? C(12,10) = C(12,2) = 66

  14. How many sets are there of three distinct integers between 1 and 50 inclusive whose sum is even? C(25,3) + C(25,1)*C(25,2) = 9800