Mathematics
MATH 3013 – Discrete Mathematics
Directions:
Show the significant steps of your work clearly for ALL problems. You may receive zero or reduced points for insufficient work.
1. Find an explicit formula for the recurrence relation
(a) an = −3an−1 + 11an−2 + 3an−3 + 10an−4
(b) an = 13an−1 − 57an−2 + 99an−3 − 54an−4
2. Solve the recurrence relation subject to the initial conditions
(a) wn = −10wn−1−25wn−2, and w0 = 5, w1 = −30
(b) −3sn = 4sn−1 − 4sn−2, and s0 = −1, s1 = 2
3. Assume that (4 − t2)Σ∞ n=2n(n − 1)antn−2 = −2Σ∞ n=0antn for all t.
(a) Show that the coefficients an is given by the recurrence relation an+2 = −(n − 2)an 4(n + 2) , for n ≥ 0.
(b) If a0 = −2, a1 = 6, find a2, a3, and a4
4. The options available on a particular model of a car are four interior colors, seven exterior colors, three types of seats, five types of engines, and two types of radios. How many different possibilities are available to the customer?
5. How many different car licensed plates can be constructed if the licenses contain four letters followed by three digits if
(a) repetitions are allowed?
(b) repetitions are not allowed?
6. How many strings can be formed by ordering the letters ”SUBBOOKKEEPER”.
7. Two dice are rolled simultaneously. How many out- comes give a sum
(a) of 2?
(b) less than 9?
(c) greater than or equal to 5?
8. In how many ways can we select a committee of three men and five women from a group of seven distinct men and nine distinct women.?