# Homework #7

Due in lecture, Thursday May 9 (with homework #8). Please do the homework in a workbook.

In all cases, you can leave your answer in terms of factorials or $$C(n,r)$$. No need to calculate: just simplify as much as you can with algebra.

# From the Text

Complete the following questions from the text:

• Section 5.1: 14, 28, 42
• Section 5.2: 12, 40, 42
• Section 5.3: 16, 24, 32, 40
• Section 5.4: 22
• For 22(b) what they mean is “use algebra”.

# Questions

1. Most bank cards have four digit identifiation numbers (PINs). Each of the four can usually be any one of 10 digits. Of course, banks want to have a lot of different PINs, but keep them short enough that people don't forget theirs. (Mine is 1234, so I don't forget it.) How many different PIN numbers are there if a bank allows…
1. the standard 4 digits with no restrictions?
2. four digits, but no repetition of digits is allowed?
3. six digits, but the digits must be unique and in ascending order. (e.g. “123456” is okay, but “123455” and “123465” are not)
4. if the bank allows four digits, but by the use of a thermal imaging camera, you find out what the four (unique) digits of your friend's PIN are? (You know the digits, but not their order.)
2. You and some friends go to the movies: eight people in total. You are all going to sit in the same row. How many ways are there for you to sit if…
1. there are no restrictions?
2. two people just started dating, and must sit next to each other?
3. two people had a fight on the way there and refuse to sit beside each other.
3. Use the pigeonhole principle to prove that the decimal expansion of $$n/m$$ ($$m,n$$ both positive integers) must repeat with period of no more than $$m$$. [Hint: think about division using the algorithm you learned in school. Try dividing something like 1÷7 with that algorithm. What happens to start a repetition?]