Problems

Among 7 girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all 7 girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other?

Gabby the Gnome has 3 cloaks of different colors: blue, green, and brown. He also has 5 different hats: 3 yellow and 2 red. Finally, he owns 6 different pairs of shoes: 2 yellow, and 4 red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the color of his shoes to match the color of the hat?

A palindromic number is a number that reads the same backward and forward. (For example, $13531$ is palindromic.)

a) How many $5$-digit numbers are palindromic?

b) How many $5$-digit numbers are palindromic and consist of distinct digits?

c) How many $5$-digit numbers consist of $3$ distinct digits and end with 2?

d) How many $5$-digit numbers are odd and consist of $3$ distinct digits?

$10$ 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).