Discrete Mathematics Answers

A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them.
(a) How many balls must she select (minimum) to be sure of having at least three blue balls?
(b) How many balls must she select (minimum) to be sure of having at least three balls of the same color?

Solve the following two " union " type questions:
(a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or)
(b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 36 computer science majors (including joint majors), 29 math majors (including joint majors) and 5 joint majors?

Solve the following two " union " type questions:
(a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or)
(b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 36 computer science majors (including joint majors), 29 math majors (including joint majors) and 5 joint majors?

How many strings of five uppercase English letters are there
(a) that start or end with the letters BO (in the order), if letters can be repeated? (inclusive or)
(b) that start with the letters BO (in that order), if letters can be repeated?
(c) that start and end with an X, if letters can be repeated?
(d) if letters can be repeated?

How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can start with zeros.)
(a) do not contain the same digit twice?
(b) begin and end with a 1?

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:
(a) have distinct digits.
(b) are divisible by 5 and by 7.
(c) are even.
(d) are not divisible by either 5 or 7.

(a) How many bit strings of length 8 are there?
(b) How many bit strings of length 8 or less are there?
(Count the empty string of length zero also.)
(c) How many strings of 6 lower case English letters are there that have the letter x in them somewhere? Here strings may use the same letter more than once. (Hint: It might be easier to first count the strings that don't have an x in them.)
(d) A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in 2 sizes for each sex. How many different types of this shirt are made?

Given that f(x)=8x^2+7 and g(x)=4x+4 are functions from R to R, find
(a) f∘g.
(b) g∘f.

Determine if each of the following functions from {a,b,c,d} to itself is one-to-one and/or onto.
Check ALL correct answers.
(a) f(a)=d,f(b)=a,f(c)=c,f(d)=b


A. onto.
B. neither one-to-one nor onto.
C. one-to-one.
f(a)=b,f(b)=a,f(c)=c,f(d)=d


A. one-to-one.
B. onto.
C. neither one-to-one nor onto.
f(a)=c,f(b)=d,f(c)=a


A. one-to-one.
B. onto.
C. neither one-to-one nor onto.

Find the following values.
(a) ⌊1.1⌋
(b) ⌈1.1⌉
(c) ⌊−0.1⌋
(d) ⌈−0.1⌉
(e) ⌈2.99⌉
(f) ⌈−2.99⌉
(g) ⌊1/2+⌈1/2⌉⌋
(h) ⌈⌊1/2⌋+⌈1/2⌉+1/2⌉