Combinatorics | Number Theory Answers

Q. If (a,b)=1 then prove that
(a-b, a+b, ab)=1

If the natural numbers are written as a sequence 12345678910111213141516...... then the 2007th digit is ????

A student has to take 8 courses to graduate. If none of the courses are prerequisites for others, how many groups of 3 courses can the student select for the coming semester?

Select one:

a. 11
b. 12
c. 24
d. 48
e. 56

Let p be a prime number. If p divides a^2,proove that p divides a,where a is an positive integer

Seven Matrons assembled for a meeting, shake hands with one another. How many handshakes take place……………….?

there are 5 students of department A and 5 of department B.how many arrangement are possible around a circular table where 2 students of department B will not sit beside?

i solve this like this.

students of department B can sit at the middle of 2 students of department A. so there are 5 empty places between the students of department A. so in this 5 place 5 students of department B can sit in 5! Ways.
and the students of department A can change their position in (5-1)! =4! ways.

so in total it is 5!*4!. that's it. my question is we know in circular permutation its (n-1)!. so i do (5-1)! for department A students. why it is not done for department B students. they also in a circle. please explain

3 buses have to cross a long bridge, and it is so narrow that a bus cannot overtake another. The buses can
have only integer-number-velocities ranging from 10ms-1
to 20ms-1
and no two buses can have the same
speed. How many ways the velocities can be distributed among the buses so that they can cross the
bridge without any accident?

Seven bandits collected a number of gold coins. They are very wary about
dividing the coins. After dividing the coins between them the first time, they found
2 coins remaining. They fought about these coins and 3 bandits died. They divided
the coins again and saw that 2 coin remains. Again they fought and another bandit
died. Now they could equally distribute the coins. At least how many gold coins
were there?

Now you are in 1st floor. You have to go 6th floor. You are able to cross three
floors at a stretch. Then you need to eat a chocolate. There are 3 chocolates for
you in 2,3 and 5th floor. How many different ways you can go 6th floor?

A street sign can be ordered with the following options:

Sheeting: Option A or Option B
Power Type: Option A or Option B
Timer: Option A, Option B or Option C
Push Button: Option A, Option B or Option C
Channel: Option A, Option B or Option C
Sensor: Option A, Option B or Option C

How many total configurations are there?