Double languages
Amont $val6 pupils of a class, $val7 speak French, $val8 speak German. (Each pupil speaks either French or German.) Now we have to choose two among these $val6 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 $val6 aligned lamps. In order to save energy, the hotel turns off $val7 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 $val7 lamps?
Computer room I
A school has a computer room with $val6 PC's. A group of $val7 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 $val7 PC's. A group of $val8 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 $val6 pupils?
Triangles in polygon
Let P be a regular polygon of $val6 sides. How many distinct triangles are there, whose 3 vertices are vertices of P?
Letterboxes
How many different ways there are, to put $val6 letters into $val7 letterboxes?
Sweets
How many ways there are to distribute $val15 sweets to $val6 girls and $val7 boys, so that $val13?
bus and drivers
There are $val6 bus, $val6 drivers et $val6 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 $val6 boys and $val7 girls. How many possibilities are there to compose a committee of class with $val14 puples, knowing that there must be at least $val9 $val11 and $val10 $val12 in the committee?
Couples
How many possibilities are there to form $val6 couples from $val7 men and $val8 women?
Groups of pupils
In how many ways can one divide a class of $val9 pupils into $val8 groups of $val7 pupils each?
Helico
We have $val6 tourism helicopters, $val6 pilots and $val7 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 $val6 lines?
Intersection points II
In the plane, two lines has at most one intersection point. How many intersection points there are at most among $val6 lines, of which $val7 are $val8 (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 $val6 lines, of which $val7 contains the origin of the plane?
Monomial 3
How many integers are there of the form $val7a·$val8b·$val9c, where the exponents a,b,c are non-negative integers with a+b+c = $val10?
Monomial 4
How many integers are there of the form $val7a·$val8b·$val9c·$val10d, where the exponents a,b,c,d are non-negative integers with a+b+c+d = $val11?
Words
How many distinct words can be formed from the $val6 first letters of the alphabet, knowing that each letter is used exactly once in each word, and that the $val7 first letters {$val8} must appear grouped in each word?
Binomial coefficients
Let n be a positive integer such that Cn$val7=Cn$val8. $val12
Binomial coefficients II
If Cn$val7=$val8, What is the value of n ?
Fixed partitions
In how many ways can we write $val9 = n1+n2+...+n$val7 , where the ni are integers greater than or equal to $val8, arranged in $val10 order?
Handshakes
$val6 couples and $val7 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 $val6 positive integers and $val7 negative integers. The absolute values of these $val8 integers are distinct prime numbers. What is the number of $val11 of two different numbers in S?
Quadrilaterals on lines
We have 2 parallel lines in the plane. On the first line we have $val6 points, and on the second, $val7 points. How many quadrilaterals can be formed by these $val8 points?
Rectangles
We have $val8 lines in the plane, of which $val6 are horizontal, and $val7 are vertical. How many rectangles are formed by these $val8 lines?
Subsets
A finite set S has $val8 subsets of $val7 elements. What is the number of elements of S ?
Dinner table
$val6 couples take dinner together. How many ways there are for these $val7 people to sit around the table, such that each gentleman is between two ladies?
Dinner table
$val6 couples take dinner together. How many ways there are for these $val7 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
$val6 couples take dinner together. How many ways there are for these $val7 people to sit around the table, such that each husband is at the side of his wife?
Dinner table
$val6 couples take dinner together. How many ways there are for these $val7 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 $val6 lines in the plane, of which $val7 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 $val6 lines?
Triangles on lines
We have 2 parallel lines in the plane. On the first line we have $val6 points, and on the second, $val7 points. How many triangles can be formed by these $val8 points?