Permutations, Combinations and Binomial Theorem Review

1. On a quiz there are 5 multiple-choice questions each with 4 choices. How many ways of answering the questions are possible?

2. A restaurant’s menu shows a choice of 6 different dinners, 4 desserts and 5 beverages. In how many ways can a person order a meal if a dinner, dessert, and a beverage is ordered?

3. Simplify:

a b =

c. In Pascal’s Triangle, what is

4. Solve for n:a without a formula, find

b

c d.

LEAVE ANSWERS IN UNSIMPLIFIED FACTORIAL FORM:5. Using the letters of the word INTERMITTENT how many 12-letter words can be

formed:a. without restrictions

b. if each word begins with T and ends with N

c. if the 4 T’s are together

d. if the T’s are in their original position

e. if each word begins with 2 T’s

f. if each word begins with exactly 2 T’s

6. Using the letters of the word LOGARITHM, how many 9-letter words can be formed:a. without restrictions

b. if the letter L must be in the middle

c. if the order LGR must be the same

d. if the letters O, A, and I must remain in the exact same position

e. if the letters L , O , and G are not together

7. How many ways can 4 boys and 4 girls be arranged alternately in a row if:a. there are no other restrictions

b. if a boy must be first in the row.

8. A construction camp is equipped with 5 signal flags of different colors. How many different signals can be sent if at most 3 flags can be flown from the flagpole at a time?

9. How many ways can the letters of the word ENGLISH be arranged if the order of the vowels are not changed?

10. Six identical math texts, a chemistry, a physics, and a biology text are arranged on a shelf. How many arrangements:a. are there

b. will not have all of the math books together

c. will have the chemistry and physics text adjacent

d. will have the chemistry text at the front and the biology text at the back

11. Using only 877799, how many even 6 digit numbers may be formed?

12. A dance committee consists of 4 girls and 5 boys. How many decorating committees of 6 can be formed if a committee is to have a girl as chairman, a boy as lighting director, and exactly 3 girls on the committee?

13. From 8 men and 5 women, a committee of 4 is selected. How many groups will havea 3 women and 1 man

b. Mary but not Roger

c. no more than 2 men

d. Roger and Joe

e. at least 1 of each gender

f. Roger or Joe

14. Points are marked on the circumference of a circle. If 20 different triangles can be formed using these points as vertices, what is the minimum number of points that are marked?

15. How many different four-letter words can be formed by using 2 letters from GLAZED and 2 from FROM?

16. A box contains 5 red, 3 green, and 6 blue marbles. Three marbles are selected at random without replacement. Determine the number of ways of selecting:

a. exactly 2 blueb. at least 1 is bluec. all 3 are of the same color

17. a. If the number of terms in the expansion is 16, then n is

b. The value of n in the expansion of if the 6th term is 38 760 is

A

B

18. Given find

a. the general term in simplified form

b. the coefficient of

c. the middle term

d. the number of the term containing

e. the constant term

19. Give the last three terms of the expansion of

20. Find the ninth term of . How many terms would be in the expansion?

21. On each grid, how many different paths are there from A to B in the following street arrangements.

a. b.

c. d

A

B

C

A

B

e.

f. You must go through B to get to C g.

Determine the number of distinct pathways from A to B assuming you do not backtrack.

h. i.

22. Consider the first part of one row in Pascal’s triangle:

a. Part of which row in the triangle is illustrated?

b. List all the other numbers in the row explaining how you determined their value.

A B

B

A

c. Explain your strategy for determining the value of the numbers in the row above this row.

23. A regular polygon has 170 diagonals. How many sides does it have?

24. A pizza shop has 5 favorite toppings. The number of different ways of selecting one or more of these toppings is

25. Suppose there are a certain number of points on a circle. If 120 triangles are formed by joining any 3 of these points, how many points are on the circle?

26. A club contains 15 members. In how many ways coulda. an executive of 3 members be selected?

b. a president, vice-president, and treasurer be selected?

Probability with Permutations and Combinations

27. In the game of cribbage, 6 cards are dealt. What is the probability ofa. being dealt four 5’s b. at least one 5

28. A lock on a briefcase has 3 wheels, each labeled from 0 to 9. What is the probability of a person’s guessing the correct entry code, if

a. there are no repeated digits in the code

b. the code may have repeated digits

29. A collection of 15 transistors contains 3 that are defective.If 2 transistors are selected in succession without replacement, what is

the probability that:a. at least 1 of them is non-defective

b. 1 defective and 1 non-defective

c. neither are defective

d. the first is defective and the second non-defective.

30. In a swim meet, there are 8 entries, 3 of whom come from the Coronation Swim Club. If we assume that their abilities are about the same, what is the probability that:a. the Coronation Swim Club, Anna, Beth and Candice, will finish 1st, 2nd, and 3rd.

b. there will be no Coronation swimmers in the top 331. A bag contains 7 red and 4 white marbles. Three balls are selected. What is the probability that the following occurs?a. all the same color b. at least 1 white

KEY SOLUTIONS1. 2.

3. a. b. c.

4. a. b. c. d.

5. a. b. c.

d. e. f.

6. a. b. c.

d. e.7. a. b.8.

9. a. 10. a. b.

10. c. d.

11. 12.

13. a. b.c. d.e. f.

14. 15.

16. a. b. or

c.17. a. 6 b. 2018. a. b. c. 252

d. 8th term e. 25219.20. , 12 terms (i.e. degree + 1)21. a. 60 b. 48 c. d.

e. 30 f. 200 g. 322 h. 18 i. 44 10022. a. 19th row

b. since rows are symmetrical about its middle value and 43 758 has been repeated; therefore the remaining numbers are 31 824, 18 564, 8 568, 3 060, 816, 153, 18 and 1.

c. The value of a number in a row below a given row can be determined by adding the two numbers immediately above it, we reverse this process to determine the values in the row above this row. E.g. The first number must be 1 and the therefore the next number must be 17 since 1 + 17 = 18.

23. 24. 31 25. 10 26. a.26. b. 27. a. 0.000 055 b. 0.397 23

28. a. 0.001 389 b.

29. a. b. c. d.

30. a. b. 31. a. b.

Written Response Questions Name ________________1. Use the expansions of for each of the following

Find the 6th term

Find the coefficient of the middle term

Find the number of ways of choosing 2 or more people from 6 people using choose notation.

State how you could use and Pascal’s triangles to solve the question in bullet 3

2. Use the expansion of for each of the following.

Find the 5th term

Find the coefficient of the 8th term

Find the term containing

Find the constant term

3. Use the letters in the word ENGINEER for each of the following.

Find the number of arrangements of all the letters.

How many arrangements begin with the letter g?

How any arrangements have the g and r adjacent and the g before the r?

How many arrangements have the 3 E’s together?

How many arrangements begin and end with an N?

4. Use the following information for the next questions.

The video store that Joe visits has 6 video games and 5 movies that he likes. He has enough

money to buy a total of 5 items ( movies and video games together).

gr eninee

ee

gninr

Joe realizes he has enough money to purchase a total of 5 items and will select from the ones he likes. How many different selections can he make?

How many ways can he select 2 video games and 3 movies?

How many ways can be select at least one movie and the rest video games?

Written Response Questions 1. Use the expansions of for each of the following

Find the 6th term

Find the coefficient of the middle term

Find the number of ways of choosing 2 or more people from 6 people using choose notation.

State how you could use and Pascal’s triangles to solve the question in bullet 3

2. Use the expansion of for each of the following.

Find the 5th term

Find the coefficient of the 8th term

Find the term containing

Find the constant term

3. Use the letters in the word ENGINEER for each of the following.

Find the number of arrangements of all the letters.

How many arrangements begin with the letter g?

How any arrangements have the g and r adjacent and the g before the r?

6 x 5 x 4x 3 x 2 x 1gr is a unit and can be located in 7 positions therefore

How many arrangements have the 3 E’s together?

5 x 4x 3 x 2 x 1eee’s is a unit and can be located in 6 positions therefore:

gr eninee

ee

gninr

How many arrangements begin and end with an N?

N __ __ __ __ __ __ N

5. Use the following information for the next questions.

Joe realizes he has enough money to purchase a total of 5 items and will select from the ones he likes. How many different selections can he make?

How many ways can he select 2 video games and 3 movies?

How many ways can be select at least one movie and the rest video games?

The video store that Joe visits has 6 video games and 5 movies that he likes. He has enough

money to buy a total of 5 items ( movies and video games together).