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Enumerative Combinatorics

  1. Apr 15, 2010 #1
    1. (a) How many repeating three-digit numbers can be formed using the numbers {1, 2, 3}?
    (b) How many non-repeating three-digit numbers can be formed using the numbers {1, 2, 3}?

    My solutions:
    a. 3 x 3 x 3=27
    b. 3 x 2 x 1 =6

    2. (a) Construct a tree diagram for 4 tosses of a fair coin.
    (b) How many ways can you get more than three tails?
    (c) How many ways can you get fewer than three tails?
    (d) How many ways can you get at least three tails?
    (e) How many ways can you get no more than three tails?

    My solutions:
    b. 1
    c. 12
    d. 5
    e. 15

    3. Determine the number of possible ways to mark your answer sheet for each test:
    (a) a six question true-or-false test.

    My solutions:

    a. 6 questions and two choices so 2 x 2 x 2 x 2 x 2 x 2= 64

    6. Decide whether a permuation or a combination is needed:
    (a) a telephone number
    (b) a Social Security number
    (c) a hand of cards in poker
    (d) the combination on a student gym locker

    My solutions:
    a. permutation
    b. permutation
    c. combination
    d. permutation

    7. How many different ways could first, second, and third place finishers occur in a race with
    12 runners competing?

    My solution: 1st 2nd 3rd, 12 runners
    3 ) = 220 ways
    n!/k! (n-k)!

    8. An ATM requires a four-digit PIN number using the digits 0-9. How many such PINs
    have no repeated digits?

    My solution:
    nPk= n!/n-k
    10 numbers, 4 digits
    4)= 5040
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Apr 15, 2010 #2
    Hm, seems you might need to check 2c: there are sixteen possible outcomes; four different ways to have three tails, and one possible way to have four tails. Right?

    I mean, clearly 2c + 2d = 16, so one of those answers must be wrong.
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