Problems

In the following grid, how many different ways are there of getting from the bottom left triangle to the bottom right triangle? You must only go from between triangles that share an edge and you can visit each triangle at most once. (You don’t have to visit all of the triangles.)

Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).

Mr Smith has seven children. He wants to send three of them to run some errands on a Saturday. We will send the first child at 1pm, the second child at 2pm and the third one at 4 pm. In how many ways can he choose them?

Katie is making a bouquet. She has \(12\) different flowers available, but wants her bouquet to be composed of exactly \(5\) of them. The order of the flowers in the bouquet doesn’t matter. In how many ways can she do this?

We have \(6\) digits available: \(1,3,4,5,7\) and \(9\). We wish to make a \(3\)-digit number with different digits but only using these \(6\) digits. How many ways are there of doing this? What if we want the digits within the number to be arranged in an ascending order - how many numbers are left?

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?