**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?**

**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?**

**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.**

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?**

**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?**

**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?**

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

**Qs:
**

**(a) A multiple-choice test contains 10 questions. There are four possible answers to each question. How many ways can a student answer the questions on the test if the student answers every question?
**

**(b) In the scenario of (a): How many ways can a student answer the questions on the test if the student can leave answers blank?**

**Q. 2-Part Questions!
(a) How many different three-letter initials can people have?
(b) How many different three-letter initials with none of the letters repeated can
people have?**

**Q: We have k distinct balls and n distinct bins. We want to distribute the balls among the bins. How many distributions are possible if there is no bound on the number of balls in each bin? Assume that all balls are distributed among the bins.**

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