Combinatorics | Number Theory Answers

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}?

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.)

All the five questions in an examination must be answered by a student. If it is believed that the order in which the questions are answered has impacts on the performance of the student. In how many different orders can the questions be answered
a.120
b.160
c.100
d.140

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

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

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

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

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

How many 4-letter words can be formed from the letters A, B, C, D, E, F if the first letter must not be a vowel and repetition of letters is not allowed?
a.240
b.300
c.360
d.200