OEF combinatorics --- Introduction ---

This module contains for the time being 32 numerical exercises on elementary combinatorics.

Double languages

Amont pupils of a class, speak French, speak German. (Each pupil speaks either French or German.)

Now we have to choose two among these pupils, such that one speaks French, the other speaks German. How many possible choices are there?

Lamps in a hotel

A hotel has a long corridor lit by aligned lamps. In order to save energy, the hotel turns off of these lamps during the night.

To have a minimum of lighting, one cannot turn off adjacent lamps, nor lamps at the extremities of the corridor. In this case, how many different ways there are to turn off lamps?

Computer room I

A school has a computer room with PC's. A group of pupils take a lesson in this computer room. How many different ways there are to attribute each pupil to a (distinct) PC?

Computer room II

A school has a computer room with PC's. A group of pupils take a lesson in this computer room. How many different ways there are to distribute the pupils over the PC's, such that each PC receives pupils?

Triangles in polygon

Let P be a regular polygon of sides. How many distinct triangles are there, whose 3 vertices are vertices of P?

Letterboxes

How many different ways there are, to put letters into letterboxes?

Sweets

How many ways there are to distribute sweets to girls and boys, so that ?

bus and drivers

There are bus, drivers et controllers. How many ways there are to attribute the drivers and controllers to the bus, such that each bus has one driver and one controller?

Commitee of class

We have a class of boys and girls. How many possibilities are there to compose a committee of class with puples, knowing that there must be at least and in the committee?

Couples

How many possibilities are there to form couples from men and women?

Groups of pupils

In how many ways can one divide a class of pupils into groups of pupils each?

Helico

We have tourism helicopters, pilots and hostesses. How many different ways there are to attribute the pilots and hostesses to the helicopters, such that each helicopter has one pilot and two hostesses?

Intersection points

In the plane, two lines has at most one intersection point.

How many intersection points there are at most among lines?

Intersection points II

In the plane, two lines has at most one intersection point.

How many intersection points there are at most among lines, of which are (therefore parallel)?

Intersection points III

In the plane, two lines has at most one intersection point.

How many intersection points there are at most among lines, of which contains the origin of the plane?

Monomial 3

How many integers are there of the form a·b·c, where the exponents a,b,c are non-negative integers with a+b+c = ?

Monomial 4

How many integers are there of the form a·b·c·d, where the exponents a,b,c,d are non-negative integers with a+b+c+d = ?

Words

How many distinct words can be formed from the first letters of the alphabet, knowing that each letter is used exactly once in each word, and that the first letters {} must appear grouped in each word?

Binomial coefficients

Let n be a positive integer such that Cn=Cn.

Binomial coefficients II

If Cn=, What is the value of n ?

Fixed partitions

In how many ways can we write

 = n1+n2+...+n ,

where the ni are integers greater than or equal to , arranged in order?

Handshakes

couples and non-maried people meet during a party. At the start of the party, each participant shake hands with each other participant once, except that there is no handshake between husband and wife. How many handshakes have occurred in total?

Positive negative

Let S be a set containing positive integers and negative integers. The absolute values of these integers are distinct prime numbers.

What is the number of of two different numbers in S?

Quadrilaterals on lines

We have 2 parallel lines in the plane. On the first line we have points, and on the second, points. How many quadrilaterals can be formed by these points?

Rectangles

We have lines in the plane, of which are horizontal, and are vertical.

How many rectangles are formed by these lines?

Subsets

A finite set S has subsets of elements. What is the number of elements of S ?

Dinner table

couples take dinner together. How many ways there are for these people to sit around the table, such that each gentleman is between two ladies?

Dinner table

couples take dinner together. How many ways there are for these people to sit around the table, such that each gentleman is between two ladies, and each husband is at the side of his wife?

Dinner table

couples take dinner together. How many ways there are for these people to sit around the table, such that each husband is at the side of his wife?

Dinner table

couples take dinner together. How many ways there are for these people to sit around the table, such that each gentleman is between two ladies, and that no husband is at the side of his wife?

Triangles

We have lines in the plane, of which contains the origin of the plane. There is no other point contained in more than two lines, nor any line parallel to another.

How many triangles are formed by these lines?

Triangles on lines

We have 2 parallel lines in the plane. On the first line we have points, and on the second, points. How many triangles can be formed by these points?
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