For all problems, respond with an integer, not a formula. Draw a box around each answer. Show your
work. Calculators are permitted.
1. A machine has a line of 9 dials, each with five settings labelled 0, 1, 2, 3, and 4. How many of
different settings of the dials are there, if no two adjacent dials have the same setting?
2. In how many ways can the letters in PANDEMICS be arranged with exactly two consecutive
vowels?
3. Four distinct numbers are simultaneously chosen from the following list:
-5, -4, -3, -2, -1, 1, 2, 3, 4, 5. (a) In how many ways can this choice be made so that the product
of the four numbers is positive? (b) In how many ways can this choice be made so that the
product of the four numbers is negative?
4. In how many ways can 17 be written as the sum of 2's and 3's if the order of the summands is
(a) not relevant, (b) relevant?
5. In how many ways can one travel in the x-y plane from (1, 2) to (5, 9) if each move is one of the
following types:
\( (R) \quad (x, y) \to (x+1, y) \) \( \quad \text{(a rightward move)} \)
\( (U) \quad (x, y) \to (x, y+1) \) \( \quad \text{(an upward move)} \)
\( (D) \quad (x, y) \to (x+1, y+1) \) \( \quad \text{(a diagonal move)} \)
