Problems

David has $15$ video games in boxes on his shelf. His family is visiting his aunt next week. He was asked to pick only $4$ games to play on his cousin’s computer. In how many ways can he do this?

Katie is making a bouquet again. She has $12$ flowers, but this time she wants to use not $5$, but $7$ flowers for a bouquet. In how many ways can she do this? How is this answer related to the answer to the previous question about Katie? Why?

Rithika is choosing songs for a party tonight. She has $214$ songs in her library and wants to use $50$ for the party. She wants to play each song only once. In how many ways can she compose her playlist?
Now suppose that each song has a different duration, and Rithika wants the songs to play in order from the longest chosen to the shortest chosen. How many ways can she choose her playlist now? (You can leave the answer as a formula).

We wish to lock a vault with different locks. The vault committee has $11$ members, each of whom has keys to some of the locks, but not all of them.

What is the smallest possible number of locks that we need to lock the vault so that each group of $6$ members can open it together with the keys they have, but no group of just $5$ members can ever do it? Note that a lock can have multiple keys that open it and a person can have keys to more than one lock.

Tommy has written 6 letters and addressed 6 envelopes. He then forgot which letter goes where and put them randomly such that no letter goes in the right envelope. In how many ways can he do this?

Annie and Hanna are preparing some Christmas baubles. They want to paint each bauble all in one colour. They have $7$ different colours of paint and $26$ baubles to paint. In how many ways can they do this? Two ways are considered the same if the numbers of baubles of each colour are the same. Each bauble has to be painted but not all the colours need to be used.

An $8\times 8$ square is divided into $1\times 1$ cells. It is covered with right-angled isosceles triangles (two triangles cover one cell). There are 64 black and 64 white triangles. We consider "regular" coverings - such that every two triangles having a common side are of a different colour. How many "regular" covers are there?

You are given a pentagon $ABCDE$ such that $AB=BC=CD=DE$, and $\mathrm{\angle }B=\mathrm{\angle }D={90}^{\circ }$. Show how the plane can be tiled with pentagons congruent to the given one.

Ms Jones vacuums her car every 2 days, she washes her car every 7 days and polishes it every 52 days. The last time she did all three types of cleaning on one day was on the 13th of March last year. What time will she do it again?

The numbers $a$ and $b$ are integers and $a>b$. Show that the gcd of $a$ and $b$ is equal to the gcd of $b$ and $a-b$.