UCL Online STEP and AEA Preparation Sessions
Session 4: Counting and Placement
Combinatorics is a branch of mathematics partly concerning the study of finite collections of objects. Aspects of combinatorics include counting the number of objects of a given kind or size, deciding when the objects can be arranged so that certain criteria can be met; and finding the "largest", "smallest", or "optimal" objects. For example
How many three digit numbers are there whose digits sum to
25?
Permutations and combinations, both of which involve working with factorial expressions, can play a central role in combinatorics problems.
A placement problem is one where objects (often numbers) have to be placed in certain positions subject to certain conditions. A magic square is a good example of this:
Show that this is the only (apart from rotating and reflecting) magic square (the sum of all the rows, columns and the two diagonals is the same) containing the numbers 1 – 9 inclusive.
To solve placement problems you usually have to ask yourself lots of ‘what if’ questions such as ‘what would be the consequence of placing 5 in the second box?’
Frequently in problems like this, you find that that a consequence is something that can’t be done and therefore you can rule out possibilities this way.
Because you often solve placement problems this way and the deductions can get quite complex, it’s a very good idea to check that any proposed complete solution you come up with actually works!