Homework 4: Counting and Discrete Probability

Note: ^ means "raised to the power."

1. (2 points) In the English alphabet using only uppercase letters, how many different three-letter initials with none of the

letters repeated can people have?

*a. 15,600

b. 17,576

c. 78

d. Around 4.03e26.

2. (2 points) The total number of strings of four decimal digits have exactly three digits that are 3s is...?

*a. 36

b. 12

c. 9

d. 10

3. (4 points) . Show that in a group of 10 people (where any two people

are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends.

4. (2 points) Mathematically speaking, a "random variable" is a:

a. variable

b. derivative

c. constant

*d. function

5. (2 points) The following statement is TRUE for what value of n?

Whenever n girls and n boys are seated around a circular table there is always a girl both of whose neighbors are boys.

a. 16

*b. 17

c. 18

d. 20

6. (2 points) What is the minimum number of integers among any group of 6 integers (not necessarily consecutive), that when divided by 5 will have the same remainder?

a. 3

b. 6

c. 5

*d. 2

7. (2 points) How many possibilities are there for the first, second, and third positions in a car race with 15 cars if all orders of finish are possible?

a. 3375

b. 3027

*c. 2730

d. 3360

8. (1 point) How many terms are there in the expansion of (x + y)^10 after like terms are collected?

a. 10

*b. 11

c. 5

d. 12

9. (2 points) If a menu at a banquet dinner has four possibilities for an appetizer, three for an entree, and five for dessert, how many possible meals are there?

*a. 60

b. 12

c. 72

d. 144

10. (2 points) You have a dozen eggs in your refrigerator, and three of them are rotten. If you reach in and pull out one at random, what is the probability that you have a rotten egg?

a. .3

*b. .25

c. .12

d .4

11. (2 points) If we have 7 poker players to seat at a 7-sided table, but all we care about is who is next to who, not which exact seat any player is in, the number of possibilities that concern us is:

a. 5040

b. 120

*c. 720

d. 840

12. (2 points) How many integers from 1 to 100 are multiples of 2 or 5?

a. 70

*b. 60

c. 65

d. 55

13. (2 points) There are 4 available flights from New York to London and, regardless of which of these flights is taken, there are 8 available flights from London to Paris. In how many ways can a person fly from New York to London to Paris?

a. 8

b. 16

c. 24

*d. 32

14. (2 points)  A class has 30 students enrolled. In how many ways can all 30 be put in two rows of 15 each (that is, a front row and a back row) for a picture?

a. 15!

*b. 30!

c. 15

d. 30

15. (2 points) If there are 15 children in a family, on how many days of the week were at least 3 children born?

a. 1

*b. at least 1

c. 2

d. at least 2

16. (2 points) Find the tenth term in the expansion of (a + b)^12.

a. 220a^6b^6

b. 66a^2b^10

*c. 220a^3b^9

d. 12ab^11

17. (4 points) Explain why an entry in Pascal's Triangle is the sum of the two entries diagonally above it.

18. (2 points) The expansion of (a + b)^7 is:

a. x^7 + 7x^6y + 14x^5y^2 + 35x^4y^3 + 35x^3y^4 + 14x^2y^5 + 7xy^6 + y^7

b. x^7 + 7x^6y + 21x^5y^2 + 28x^4y^3 + 28x^3y^4 + 21x^2y^5 + 7xy^6 + y^7

c. x^7 + 7x^6y + 21x^5y^2 + 42x^4y^3 + 42x^3y^4 + 21x^2y^5 + 7xy^6 + y^7

*d. x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7