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?