Permutations I
1. Find the number of ways that digits 1 through 6 can be used to form a 4-digit number. Each digit can be used at most once.
2. Find the number of ways that digits 0 through 5 can be used to form a 4 digit number (ie 1000 or over). Each digit can be used at most once.
Hint 1
Identify any digits different from the others, and any place value different from the others. Account for these differences.
3. A certain set of cards has 5 colors each numbered 1 to 8 for a total of 40 cards. How many ways can two cards be taken out without replacement such that the first card is of a lower number than the second?
Hint 1
Find and apply symmetry.4. A cinema has 13 seats in a row. There are three groups of 2, a group of 3, and 4 others (who are not part of any groups) to be seated in this row. Everyone in each group must be seated together. Find the number of ways to seat all 13 visitors.
Hint 1
Each group can be arranged independent of where the group is and how other groups are rearranged.
Hint 2
Multiply the ways to place each group (and persons not part of any group) by the ways to permute the persons in each group.
5. Assume there are 365 days in a year and people are equally likely to be born on any given day. Find the minimum number of people in a group so that there is at least 0.75 chance of having at least two people sharing the same birthday.
Hint 1
Consider the complement of people have all distinct birthdays.
Hint 2
This is known as the birthday problem.