\(10\) people including Alice, Bob and Charlie are waiting in a queue. How many distinct line-ups are there such that none of the mentioned three are next in the queue to both of the other two?
Mollie’s mum would like to buy 16 balloons. The balloons come in three colours: red, green, and blue. In how many ways can she buy these balloons if she would like to get at least 4 of every colour?
Mr. and Mrs. Jones have six kids – 3 boys and 3 girls. Today, a photographer is taking pictures of the family.
a) In how many ways can the kids be seated in a row so that all the girls are on the left and all the boys are on the right?
b) In how many ways can Jones’ kids be seated in a row so that girls and boys alternate?
c) In how many ways can the whole family be seated, if all the girls must be sitting together, all the boys must be sitting together as well, and parents must be either together in the centre, or on both sides?
How many 12-digit numbers, whose product of digits equals 6, are there?
Matt has a cube and wants to colour each face a different colour. He has \(6\) dyes prepared. In how many different ways can he do it? Two colourings are different if the cube cannot be rotated to look like the other one.
It is happy hour on Friday. Sue, Sam, Pete, Martha and Bradan are fooling around at their office desks. There are \(5\) desks, which correspond to where they sit during the day. How many ways are there for them to occupy a seat at the various desks, such that nobody is in the correct spot?
WizardLand Middle School offers a new elective this year: an astrology class. Fifteen students have registered for this course. Prove that at least 2 of these students were born under the same zodiac sign (there are 12 zodiac signs in total, one for each month).
Leo’s dad was making a pizza for lunch. He decided to place 7 pieces of pineapple on it. Assuming the pizza is a circle of a \(20\) cm radius, show that some two pieces of pineapple were placed closer than \(20\) cm apart.
With a red marker, Margaret marked three points with integer coordinates on a number line. With a blue marker, Angelina marked a midpoint for every pair of red points. Prove that at least 1 of the blue points has an integer coordinate.
Prove that out of any 11 natural numbers, 2 can be found such that their difference is a multiple of 10.