Combinatorics | Number Theory

In a certain country, telephone numbers have 9 digits. The first two digits are the area code and are the same within a given area. The last seven digits are the local number and can’t begin with 0. How many different numbers are possible within a given area code in this country?

Combinatorics | Number Theory

A rectangle with height and width equal to 3 and 25 respectively, is drawn on a checkered paper. Bazil paints a random horizontal 1×2 rectangle, and Peter paints a random vertical 2×1rectangle (each rectangle consists of 2 sells). Find the probability that at least one of the cells is painted twice. Express the answer in percent, and round to the nearest integer.

Combinatorics | Number Theory

Let us consider two irreducible fractions. The denominator of the first one is equal to 8200,and the denominator of the second to 4300. What is the smallest possible denominator of a fraction equal to the sum of these fractions, after the fraction is reduced? (For example, (2/3) + (8/15) = (18/15) = (6/5), and the denominator after the reduction is equal to 5.)

Combinatorics | Number Theory

Let us denote S_n=a^n+b^n+c^n for arbitrary numbers a,b,c. It is known that S_1=4,5, S_2=22,25, S_3=104,625 for some values of a,b,c. What is the largest possible value of S_{737}^2-S_{736}S_{738}?

Combinatorics | Number Theory

37 students are members of a sports club. Every two of them are either friends or enemies.(Friendship and enmity are reciprocal, i.e. if A is a friend to B then B is a friend to A, and the same applies to being enemies.) It has turned out that each of the students has exactly 8 enemies. Let us called a group of three students concurrent if they are either pairwise enemies or pair wise friends to each other. What is the maximum possible quantity of concurrent student triples in this sports club? (Two distinct concurrent student triples may have mutual students in them.)

Combinatorics | Number Theory

Find the number of distinguishable permutations that can be formed from the letters of the word "CINCINNATI"

a.604800

b.25200

c.50400

d.100800

a.604800

b.25200

c.50400

d.100800

Combinatorics | Number Theory

The letters V, W, X, Y and Z are selected at random to form a five letters word without repetition. How many ways can a word be formed such that X, Y and Z follow one another?

a.36

b.48

c.24

d.12

a.36

b.48

c.24

d.12

Combinatorics | Number Theory

The letters V, W, X, Y and Z are selected at random to form a five letters word without repetition. How many ways can a word be formed such that X and Y follow each?

a.60

b.48

c.12

d.24

a.60

b.48

c.12

d.24

Combinatorics | Number Theory

In how many ways can 4 married couples attending a concert be seated in a row of eight seats if each couple must be seated together?

a.384

b.192

c.129

d.16

a.384

b.192

c.129

d.16

Combinatorics | Number Theory

In how many ways can 4 married couples attending a concert be seated in a row of eight seats if members of same gender must be seated together?

a.756

b.1152

c.576

d.24

a.756

b.1152

c.576

d.24