In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
A counter is placed in the bottom right hand corner of a grid. You
toss a coin and move the star according to the following rules: ...
What is the probability that you end up in the top left-hand corner
of the grid?
Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over...
You win if all your cards end up in the trays before you run out of cards in one pile. What is the probability that you win?
To answer questions like this you need to have a good way of representing reorderings. Our aim is to introduce you to a useful way of doing this called cycle notation
that will help you manipulate these permutations on paper. A permutation takes an ordered set of elements and shuffles them into a new order.
Work through the examples below, and then you should be able to solve our problem.
You'll find the shuffle interactivity
helpful. There is also a short tutorial
that explains how to create and manipulate shuffles.
The picture shows a simple shuffle which can be written as (1 4 3 2) in cycle notation. Think 1 goes to 4, 4 goes to 3, 3 goes to 2, and finally 2 goes back to 1 forming a cycle.
Not every shuffle can be represented with just one cycle. Here are two that require 3 cycles.
There is one special case - the identity shuffle which does nothing. Call this (1).
The permutation (2 3) reorders the second and third elements, leaving everything else unchanged. In the shuffles interactivity this same permutation can appear as different length shuffles like this:
Try using the interactivity to make these permutations:
Now, back to the main question:
Can you write down all the ways that you can reorder the letters in sheffuls
to make the word shuffles
using the cycle notation?