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”.
- 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…
- the standard 4 digits with no restrictions?
- four digits, but no repetition of digits is allowed?
- six digits, but the digits must be unique and in ascending order. (e.g. “123456” is okay, but “123455” and “123465” are not)
- 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.)
- 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…
- there are no restrictions?
- two people just started dating, and must sit next to each other?
- two people had a fight on the way there and refuse to sit beside each other.
- 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?]