Discrete Mathematics for Computer Science

Exercises 433

If we use the rate estimate for crypt operations of 200,000 per second, then it would

take about 15,385,647 seconds-about 4273 hours, which is almost half a year-to try all

possible passwords. U

rnExercises

1. How many license plates can be made using two uppercase letters followed by a 3-digit
   number?


2. How many ways can you draw a club or a heart from an ordinary deck of cards? A
   spade or an ace? An ace or a jack? A card numbered 3 through 9? A numbered card
   (Aces are not numbered cards) or a king?


3. How many ways can one choose one right glove and one left glove from six pairs of
   different gloves without obtaining a pair?


4. In planning a round trip from Cleveland to Dover by way of New York, a traveler
   decides to do the Cleveland-New York segments by air and the two New York-Dover
   segments by steamship. If six airlines operate flights between Cleveland and New York
   and four steamship lines operate between New York and Dover, in how many ways can
   the traveler make the round trip without using the same company twice?


5. Given the digits 1, 2, 3, 4, and 5, find how many 4-digit numbers can be formed from
   them:
   (a) If no digit may be repeated
   (b) If repetitions of a digit are allowed
   (c) If the number must be even, without any repeated digit
   (d) If the number must be even


6. How many odd numbers between 1000 and 10,000 have no digits repeated?


7. How many natural numbers greater than or equal to 1000 and less than 5400 have the
   properties:
   (a) No digit is repeated.
   (b) The digits 2 and 7 do not occur.


8. How many 7-digit numbers are there such that the digits are distinct integers taken
   from { 1, 2 ... , 9} and the integers 5 and 6 do not appear together in either order?


9. How many 6-digit numbers can be formed using { 1, 2, 3,..., 9) with no repetitions
   such that 1 and 2 do not occur in consecutive positions?


10. Answer the following questions about 9-digit natural numbers with no repeated digits
    (leading zeros are not permitted):
    (a) How many such 9-digit numbers exist?
    (b) How many are divisible by 2?
    (c) How many are divisible by 5?
    (d) How many are greater than 500,000,000?


11. How many positive integers less than 1,000,000 can be written using only the digits 7,
    8, and 9? How many using only the digits 0, 8, and 9?


12. Find the number of ways that flipping five coins can give at least three heads. Use the
    Multiplication Principle and symmetry to conclude that the answer is 24.