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Unit 3- Review
Organized Counting and Permutation

1. Some terms of a row of Pascal’s triangle are shown below.

1, 11, 55, 165, 330, 462, ……

a. Determine the row number. b. Determine how many items are in this row. c. Determine the remaining terms in this row. d. Determine the sum of the numbers in this row. e. Determine the terms of previous row.

2. Express as a single term from Pascal’s triangle. a. [pic] b. [pic] c. [pic] d. [pic] e. [pic] f. [pic]

3. Express in factorial notation a. [pic] b. [pic] c. [pic] d. [pic]

4. Simplify. a) [pic] b) [pic] c) [pic] d) [pic]

5. Snack Shack serves: egg or ham sandwiches; coffee, soft drink or milk; donuts or pie for dessert. What are the possible meals if one item is chosen from each category?

6. Use the letters in the word " square " and tell how many 6-letter arrangements, with no repetitions, are possible if the :

a. first letter is a vowel. b. vowels and consonants alternate, beginning with a consonant.

7. How many different 5-letter words can be formed from the word APPLE?

8. How many arrangements of the word ACTIVE are there if a. C,E must be together b. C,E must not be together

9. How many different six-digit numbers can be written using the following six digits: 4,4,5,5,5,7

10. How many different five-digit numbers can be written using the following six digits: 4,4,5,5,5,7

11. How many different license plates can be made is a plate consists of: a. three different letters followed by two different digits b. three letters followed by two numbers, not necessarily different c. three consecutive letters and three consecutive numbers?

12. In how many ways can 6 students be arranged in a circle?

Note: Arrangements are also often made in a circle—we no longer have a left end and a right end. Now our first element placed merely provides a point of reference instead of having n choices. Thus with n distinguishable objects we have (n-1)! arrangements instead of n!.

13. In how many ways can you roll a sum of 6 or a sum of 10 with a pair of dice?

14. How many ways are there to draw a 7 or a red king from a standard deck of 52 playing cards?

15. How many ten-digit telephone numbers are possible if the first three digits must all be different and no number may begin with 0?

16. How many six-digit even numbers less than 200 000 can be formed using all the digits 1, 1, 2, 2, 3, and 5?

17. How many permutations of the word committee begin or end with an e?

18. How many arrangements of five letters from the word certain do not contain the letter t?

19. How many arrangements of five letters from the word certain contain the letter t?

20. Find the number of permutations of all the letters of the word lancers that satisfy each of the following conditions. a. The letters can be in any order. b. The third letter must be n. c. The letters r and s must be adjacent. d. The letters r and s are separated by only one letter. e. The letters r and s must be adjacent while the letters a and n must be separated.

Answer Key:

1- a)11 b)12 c)462, 330, 165, 55, 11, 1 d)2048 e)1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

2- a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
3- a) [pic] b) [pic] c) [pic] d) [pic]
4- a) [pic] b) [pic] c) [pic]
d) [pic]

5- 12
6- a)360 b)36
8- a)240 b)480
9- 60
10- 60
11- a)1404000 b)1757600 c)192
13- 8
14- 6
15- 6480000000
16- 24
18- 720 math19- 1800
20- a)5040 b)720 c)1440 d)1200 e)960…...

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