MAT 142 Summer Session 1 ANSWERS Homework Day #14:
Name:______________________
I. The Counting Principle and Permutations
For problems #1 – 4 evaluate the following expressions both WITH and WITHOUT a calculator!
1. 4! 2.
7
P
2
3.
8
P
0
4.
3
P
3
5.
8
P
1
24 42 1 6 8
6. Explain, in words, what
8
P
0
means.
It is the number of permutations (order matters) of 8 objects taken 0 at a time. There is only 1 way to take 0 objects and that is to just NOT take them.
7. Explain, in words, what P means. It is the number of permutations (order matters) of 3 objects taken 3 at a
3 3 time. There are six ways to do this. 3 choices for the first object taken, 2 choices for the second object taken and 1 choice for the third object taken. Utilizing the Counting Principle we have 3 x 2 x 1 = 6 ways to do this.
8. To use an ATM you generally must enter a 4-digit code (using digits 0 – 9). How many four-digit codes are possible if repetition of digits is permitted?
10 x 10 x 10 x 10 = 10,000
9. To use an ATM you generally must enter a 4-digit code (using digits 0 – 9). How many four-digit codes are possible if repetition of digits is NOT permitted?
10 x 9 x 8 x 7 = 5040 Note: this is P
10 4
A license plate is to consist of 3 digits followed by 2 uppercase letters. Determine the number of different license plates possible if…..
10. repetition of numbers and letters is permitted.
10 x 10 x 10 x 26 x 26 = 676,000
11. repetition of numbers and letters is NOT permitted.
10 x 9 x 8 x 26 x 25 = 468,000 Note: This is
10
P
3

26
P
2
12. The first and second digit must be odd, and repetition is NOT permitted.
5 x 4 x 8 x 26 x 25 = 104,000
13. Determine the number of permutations of the letters of the word “EDUCATION”.
 or P
9 9
14. Determine the number of permutations of the letters of the word “DIFFERENCE”.
10!

302, 400
15. Joe’s Pizza shack sells three sizes of pizza (small, medium and large) with two different types of crust (thin and regular) and has the followings toppings as choices…Sausage, Pepperoni, Onion, Green Peppers, Mushrooms and
Tomatoes. Given this information, how many different pizzas can Joe’s Pizza shack make for their customers? Use the counting principle.
3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 384
Choose size Choose Crust then for each of the 6 possible toppings choose to have it or not (two choices per topping)

II. Combinations:
For problems # 16 - 19 evaluate the following expressions both WITH and WITHOUT a calculator!
18.
5
C
0
19.
5
C
5
16.
5
C
2
10
17.
5
C
1
5
20. Explain why, C
10 3

C
10 7
1 1
10
C
3


10!
 

10!
10 3 !3!
7!3!
10
C
7


10!
 

10!
10 7 !7!
3!7!
see they are the same !!!!!!!
21. Does P
10 3

P
10 7
? Explain why or why not.
10
P
3


10!
 

10!
10 3 !
7!
10
P
7


10!
 

10!
10 7 !
3!
see they are NOT the same !!!! in fact
10
P is 7 6 5 4 840 times bigger than
10
P
3
22. An ice cream shop has twenty different flavors. If Tammy wishes to have three different flavors in her “three scoop” sundae, how many selections are possible?
20
C
3

1140
23. In the Arizona Lottery’s “The Pick” game, a player must select 6 numbers from the numbers 1 through 41. This is a combination problem since if I select the numbers 5, 41, 14, 9, 16, 27 I would end up with the same ticket as someone who selects 16, 27, 41, 5, 14, 9. If each game costs $1, how much would a person have to spend to guarantee they would match the 6 numbered balls that get drawn?
There are
41
C
6

4, 496,388 many different tickets so a person would have to spend $4,496,388 to be sure they had a ticket that matched whatever 6 number combination was drawn.
24. In order for you to make money by spending that much money from problem 23, what TWO things would have to happen?
A. The jackpot would have to be bigger than $4,496,388
B. If multiple people had the “winning” ticket you could lose money even if the jackpot was higher. So you would
need it to be the case that IF there were multiple winners with which you had to split the Jackpot, you would
have to have your share be bigger than $4,496,388.
25. If , in order to win the top prize in “The Pick”, you had to select the 6 numbers in exactly the same order that the numbered balls were drawn, how many tickets would you have to buy to ensure you would win the jackpot?
41
P
6

3, 237,399,360
26. The powerball lottery game is another example of a combination problem (in that the order that you select your numbers does not have to match the order that the numbered balls are drawn). You must select 5 white numbers from the numbers 1 – 49 AND 1 red number (the powerball) from the numbers 1 – 42. How many different powerball tickets are possible?
49
C
5

42
C
1

80, 089,128
Optional Reading. If you would like to read a little more about Combinations and Permutations here are some web links that you may find helpful.
1. http://www.mathsisfun.com/combinatorics/combinations-permutations.html
2. http://www.omegamath.com/Data/d2.2.html
(Out of 60 students in my two classes, how many do you think read any of these?)