Principles of Counting Recitation

Week one

Your TA will work through the following problems with you.

1. How many six character passwords are there that start with two digits and end with four lowercase letters?
2. How many five or six character passwords are there that use only lowercase letters?
3. If there are 1500 teams playing in a tournament, where a team is eliminated after a round in which they lose, how many total games are played in this tournament?

Use your knowledge of the product and sum rules to answer the following questions.

1. How many ways are there for a person to have three initials?
2. How many bit strings of length seven begin with two zeros AND end with three ones?
3. How many permutations of start with and end with ?
4. How many different ways are there to choose a president, vice president and treasurer out of ten people?
5. How many ways are there to choose a committee of five people out of nine people?
6. How many permutations of the letters ABCDEFG contain
   • the string BCD?
   • the strings BA and GF?
7. If a password is made up of lowercase letters or digits, how many passwords of length six are there that contain AT LEAST two digits?
8. How many ways can a set of three positive integers less than 100 be chosen?
9. In how many ways can nine people sit around a circular table, all seats being identical?
10. If there is a room with 36 people and every person shakes hands with every other person exactly once, how many handshakes are there in all?
11. In a rectangular grid, how many paths are there to get from (0,0) to (3,3) only moving right or up?
12. A professor writes 40 discrete mathematics true/false questions. Of the statements in these questions, 17 are true. If the questions can be positioned in any order, how many different answer keys are possible?

Week two

1. How many permutations of ABCDEFG are there with E preceding G? What about with E preceding G or C preceding A?
2. How many anagrams of “banana” are there?
3. A drawer contains 5 blue socks, 4 green socks, 8 red socks, and 2 yellow socks
   • How many do you have to remove to ensure you have removed at least 2 socks of the same color?
   • How many do you have to remove to ensure you have removed at least 2 red socks?
4. A group of foreign language students was surveyed about languages they spoke. How many students spoke any of the three languages?
   • 150 spoke French
   • 200 spoke German
   • 250 spoke Spanish
   • 100 spoke both French and German
   • 75 spoke both German and Spanish
   • 100 spoke both French and Spanish
   • 50 spoke all three languages
5. Call a number “prime-looking” if it is composite but not divisibly by 2, 3, or 5. There are 25 prime numbers less than 100. How many prime-looking numbers are there less than 100?