1.a. Out of a total of 130 students, 60 are wearing hats, 51 are wearing scarves, and 30 are wearing both hats and scarves. Everyone wearing neither a hat nor a scarf is wearing gloves: (a) How many students wearing gloves? (b) How many students not wearing a sweater are wearing hats but not scarves? (c) How many students not wearing a sweater are wearing neither a hat nor a scarf?

1.b. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is multiple of 3 or 5?

1.c. In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

2.a. Prove by induction that the sum of the cubes of three consecutive integers is divisible by 9.

2.b. Two cards are drawn together from a pack of 52 cards. Determine the probability that one is a spade and one is a heart.

2.c. Three unbiased coins are tossed. What is the probability of getting at most two heads?

3.a. Solve the following recurrence relation: $a_{r}-7 a_{r}-1+10 a_{r}-2=2^{r}$ $\quad a_{1}=3, a_{2}=21$

3.b. Find the shortest path from vertex a to z in the following graph:

4.a. Functions f, g and h are defined on the set X = {1, 2, 3} as: f = {(1, 3), (2, 1), (3, 2)}; g={(1, 2), (2,3), (3,1)}; h={(1,2), (2,1), (3,3)}; (i) Find fog and gof. Are the equals? (ii) Find fogoh and fohog.

4.b. Define Isomorphic Graphs. Show that the following graphs G1 and G2 are isomorphic.

5.a. Find the minimum spanning tree and weight of it for the given graph using Kruskal's algorithm.

5.b. Define optimal tree. For the following set of weights, construct optimal binary prefix code. For each weight in the set, give corresponding prefix code: 1,4,8,9,15,25,31,37.

6.a. Find the maximum flow for the following transport network.

6.b. Find the fundamental system of cut set for the graph G shown below with respect to the spanning tree T.

7.a. Let be the set of all rational numbers other than 1. Show that with operation * defined on the set by (a * b = a + b - ab) is an Abelian group.

7.b. Let I be the set of all integers. For each of the following determine whether * is an associative operation or not: (1) a * b = max(a, b) (2) a * b = min(x + 2, b) (3) a * b = a - 2b (4) a * b = max(2a - b, 2b - a).

8.a. Let $\mathrm{Z}_{n}$ be the set of integer {0, 1, 2, ...., n - 1}. Let be a binary operation on $Z_{n}$ such that :

$a \oplus b=\left\{\begin{array}{c}{a+b i f a+b\lta} \\ {a+b-n i f a+b \geq n}\end{array}\right.$

Let be a binary operation on $Z_{n}$ such that : a b = the remainder of ab divided by n. Show that is Ring.

8.b. Consider the (2, 7) encoding function e. e(00) = 0000000, e(01) = 1010101, e(10) = 0111110, e(11) = 0110110 (a) Find the minimum distance of e. (b) How many errors will e detect?