1. A password consists of 5 characters. The first character must be a letter chosen from the English alphabet, one of the remaining 4 characters must be a digit chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and each of the remaining 3 characters must be a special character from the set {!, @, #, $, %, &}. How many different passwords are possible?

2. Find the number of ways 5 different math books, 4 different English books and 2 different biology books can be arranged in a row on a shelf

1. If the books can be arranged in any order.
2. If the math books must be next to each other.
3. If the English books must be next to each other.
4. If the math books must be next to each other and the biology books must be next to each other.
5. If the books of each subject must be next to each other.
6. If the books of each subject must be next to each other and the subjects must be in the order of math-English-biology.

3. A coin is flipped 5 times and the sequence of heads and tails is recorded. How many different sequences are possible that consist of

1. 5 heads and 0 tails?
2. 4 heads and 1 tails?
3. 3 heads and 2 tails?
4. 2 heads and 3 tails?
5. 1 heads and 4 tails?
6. 0 heads and 5 tails?

Please check that if you add the 6 numbers above together, you get 32, which comes from 2⁵. Can you see the connection?

4. The mathematical model in this problem is used in all lottery games.

There are 5 red and 4 white, numbered, marbles in a bowl, well mixed. A sample of 3 marbles is taken.

1. How many different samples are possible?
2. How many samples consist of 3 red and 0 white marbles?
3. How many samples consist of 2 red and 1 white marbles?
4. How many samples consist of 1 red and 2 white marbles?
5. How many samples consist of 0 red and 3 white marbles?
6. The sum of the four numbers from B to E = . How does it compare to A?

5. 6 members from the math department, 6 members from the biology department, and 6 members from the English department have declared themselves candidates for two vacant faculty senate seats. Suppose the two seats cannot be filled by two members from the same department, in how many ways can the two seats be filled?

6. 24 people joined a Web meeting. In how many ways can they be divided into three groups of 9, 8, and 7 people?

7. Emma has 15 different teddy bears. In how many ways can she name one bear the President, one the Vice President, one the Speaker of the Den, and 3 Senior Advisors to the President?

8. Suppose there are 13 people in a room, how many handshakes will be done if each person shakes hands with every other person exactly once?

9. There are 44 numbered balls in a drum (think lottery). A robot randomly picks 6 balls, I also randomly pick 6 balls. In how many ways can 4 of the balls picked by me be among the 6 balls picked by the robot?