Tina Evans

MTH 110     Review Worksheet for Counting Techniques and Probability with Counting Techniques  (Sections 8.1-8.4)

1)  At a lumber company that sold shelves, a customer could choose from 5 types of wood, 5 different widths and 4 different lengths. How many different types of shelves could be ordered?

2)  How many distinguishable permutations of letters are possible in the word?


3)  Four accounting majors, two economics majors, and three marketing majors have interviewed for five different positions with a large company. Find the number of different ways that five of these could be hired.  One accounting major, one economics major, and one marketing major would be hired, then the two remaining positions would be filled by any of the majors left.

4)  An order of award presentations has been devised for seven people:  Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.  In how many ways can the men be presented first and then the women?

5)  To win the World Series, a baseball team must win 4 games out of a maximum of 7 games. To solve the problem, list the possible arrangements of losses and wins.  How many ways are there of winning the World Series in exactly 6 games if the winning team wins the last two games?

6)  Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain the following cards?    Two red cards and three black cards

7)  Find the number of ways to get the following card combination from a 52-card deck.  If three cards are successively dealt from a 52-card deck without replacement, in how many ways could they be a club, then a spade, and then a heart?

8)  A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get exactly 3 apples?

9)  How many three-digit counting numbers do not contain any of the digits 1, 5, 7, 8, or 9?

10)  How many different three-number "combinations" are possible on a combination lock having 24 numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation locks.)
11)  A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committee can have at most two girls?

12)  A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability of getting 1 cherry and 2 lemon candies.

13)  Find the probability of the following card hands from a 52-card deck. In poker, aces are either high or low. A bridge hand is made up of 13 cards.    What is the probability of getting exactly 3 kings and exactly 3 queens?

14)  A family has five children. The probability of having a girl is 1/2. What is the probability of having no more than 3 boys?

15)  A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting exactly three fours.

16)  A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting more than one two.

17)  A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting exactly five twos.

18)  What is the probability that 10 rolls of a fair die will show 4 threes?

19)  A coin is biased to show 43% heads and 57% tails. The coin is tossed twice. What is the probability that the coin turns up tails on both tosses?

20)  A battery company has found that the defective rate of its batteries is .03. Each day, 22 batteries are randomly tested. On Tuesday, 1 is found to be defective. Find the probability of this event.


1)  100 types

2)  34,650

3)  720 ways

4)  144

5)  6 ways

6)  845,000 hands

7)  2197 ways

8)  30 ways

9)  100 numbers

10)  12,144 three-number "combinations"

11)  4620 ways

12)  .0364

13)  .00097

14)  .8125

15)  .032

16)  .870

17)  .129

18)  .0543

19)  32.49%

20)  .348