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1. Four different written driving tests are administered by the city. One of these tests is chosen at random for each applicant for a driver’s license. If a group of 2 women and 4 men apply, find out the number of ways:

(a) Exactly 3 of the 6 people will take the same test.

(b) The 2 women take the same test.

(c) All 4 men take different tests.

2. How many binary strings (strings of 0s and 1s) hold exactly five 0s and 14 1s if every 0 must be immediately followed by two 1s?

3. In how many ways can 6 men and 4 women sit down around the circular table?

(a) With no restrictions?

(b) If women can’t sit directly next to each other?

(c) If all 6 men should sit consecutively?

4. Consider passwords consisting of a string of 4 characters, each selected from (A, B, C, D, E, F, G).

(a) How many passwords have at least one A if repetition of characters is not allowed? An ex of such a password is BAGF.

(b) How many passwords have at least one A if repetition of characters is allowed? An ex of such a password is BAAF.

(c) How many passwords are there if consecutive characters cannot be identical, but repetition of characters is allowed? An ex of such a password is ABAF.

(d) How many passwords have exactly two As if repetition is allowed?

5. Let A = (1, 2, 3, 4, 5, 6) and B = (1, 2, 3, 4, 5, 6, 7, 8). How many binary relations from A to B contain the ordered pair (2, 5) but do not include the ordered pair (3, 3)? How many of the binary relations that you counted are also functions?

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