# Venn Diagram Word Problem

Q:  Superburger sells hamburgers with the choice of ketchup, mustard and relish. One day they sold 256 hamburgers; 140 had mustard, 140 had ketchup, 84 had ketchup and relish, 62 had mustard but no relish, 68 had ketchup and mustard, 38 had all three condiments and 20 had none.

(a) The number sold with relish only is?

(b)The number sold with no relish is?

Q:  Here are some counting problems:

(a) How may different positive integer answers are to x1 + x2 + x3 + x4 = 11?
(b) How may different positive integer answers are to x1 + x2 + x3 <= 14?

# Proof with functions

Q: Let f and g be invertible functions such that:

f: S –> T
g: T –>U
(1) Show g compose f is invertible and,

(2) that (g compose f) inverse = (f inverse) compose (g inverse)

# Discrete Logic

Q:  Show that if seven integers are chosen from the first 10 positive integers, there must be at least two pairs with the sum of 11.

# Scrambling

OK.  I must redeem myself in the realm of counting!  Too many hasty errors on my last post for discrete math and not enough “thinking.”  Don’t fall victim to this.  And now to the question:

Q:  How many bit strings (reminder: a bit is either 0 or 1) of length 10 have:

(a) Exactly 3 zeros?
(b) Have the same number of 0 and 1
(c) Have at least 7 zeros
(d) Have at least 3 ones?

# How many… part II

Q:  How many strings of four decimal digits
(a) Do not contains the same digit twice?
(b) Begin with an odd digit?
(c) Have exactly 3 digits that are 9?

# How many….

Q:  How many positive integers less than 1000 are:
(a) divisible by 7?
(b) divisible by 7 but not by 11?
(c) divisible by either 7 or 11?
(d) divisible by exactly one of 7 or 11?
(e) divisible neither by 7 or 11?
(f) have distinct digits?
(g) Have distinct digits and are even?

Q:  How many license plates can be made using either three digits followed by three letters or three letters followed by three digits?

# Ordering

Q: Eight athletes participate in a race. Calculate the number of possible outcomes of the race if no ties are allowed.