This number, using C for combination, is written^8 C 5 and isIn the UK National Lottery the rules require a selection of 6 numbers from a
possible 49 – how many possibilities are there?
Only one combination wins so there is approximately 1 chance in 14 million of
picking the jackpot.
Kirkman’s problem
Combinatorics is a wide field and, though old, it has rapidly developed over
the past 40 years, due to its relevance to computer science. Problems involving
graph theory, Latin squares and the like can be thought of as part of modern
combinatorics.
The essence of combinatorics is captured by a master of the subject, Rev.
Thomas Kirkman, working at a time when combinatorics was mostly linked to
recreational mathematics. He made many original contributions to discrete
geometry, group theory and combinatorics but never had a university
appointment. The conundrum which reinforced his reputation as a nononsense
mathematician was the one for which he will always be known. In 1850 Kirkman
introduced the ‘15 schoolgirls problem’, in which schoolgirls walk to church in 5
rows of 3 on each day of the week. If you are bored with Sudoku you might try
to solve it. We need to organize a daily schedule so that no two walk together
more than once. Using lower case and upper case deliberately, the girls are:
abigail, beatrice, constance, dorothy, emma, frances, grace, Agnes, Bernice,
Charlotte, Danielle, Edith, Florence, Gwendolyn and Victoria, labelled a, b, c, d,
e, f, g, A, B, C, D, E, F, G and V, respectively.
There are actually seven distinct solutions to Kirkman’s problem, and the one
we’ll give is ‘cyclic’ – it is generated by ‘going around’. This is where the labelling