As one does, I went into my kitchen this morning and on seeing five different cereal packets in my cupboard, suddenly wondered, how many different breakfasts could I make out of those five cereal types?

For example, if I only had one cereal a day, then I could have five different breakfasts:

Monday = Muesli
Tuesday = Bran Flakes
Wednesday = Weetabix
Thursday = Malted Shreddies
Friday = All Bran

Suppose now, though, that I mix two cereals a day: for example, Muesli and Bran Flakes … how many different breakfasts can I have now? That is, in how many ways can I combine the five different cereals, two at a time?

Make a list: and prove that the answer is that you can have ten different breakfasts from a choice of five cereals taken two at a time.

Suppose now that I make a mixture of three cereals a day from the five cereals in my cupboard: how many combinations of breakfast are available to me now?

Check that you agree that the answer has to be ten again.

Choosing four from five cereals gives me a combination of five possible breakfasts and choosing five from five cereals gives me a combination of one possible breakfast.

That’s taken a long time hasn’t it? A lot of brain work and making of lists and tables. To see a more detailed background to this topic, take a look at my web pages on combinations (and the sister idea of permutations) here:

http://www.duncanwil.co.uk/permcom.html
http://www.duncanwil.co.uk/com.html

In that first page you will learn the difference between permutations and combinations: in simple terms, permutations count ALL possible variations whereas combinations exclude equivalent outcomes.

That is, permutations would count as three different breakfasts

Muesli + All Bran + Weetabix
Weetabix + Muesli + All Bran
All Bran + Weetabix + Muesli

whereas combinations say they are the same as each other … mathematically speaking.

Excel Functions: combinations and permutations

Let’s cut this page short now and introduce two Excel functions that save us from having to create these possibly lengthy analyses to find out how many possible breakfasts I might create from my five cereals:

=COMBIN(n,r) that is, find the combinations of r breakfasts from n cereals
=COMBIN(5,3) = 10 = I can create ten different breakfasts by combining three cereals from the five cereals available

=PERMUT(n,r) that is find the permutations of r breakfasts from n cereals
=PERMUT(5,3) = 60 = I can create 60 different breakfasts by perming three cereals from the five cereals available

The differences between the number of combinations and the number of permutations can be very large and the table below shows a table to compares the combinations and permutations for this cereal and breakfast case study.

Summary    
n 5  
r Combinations Permutations
1 5 5
2 10 20
3 10 60
4 5 120
5 1 120

There you are: that’s what I thought about as I opened my cupboard to make my breakfast this morning.

Why not build the above table for yourself using =COMBIN(n,r) and =PERMUT(n.r)\; make n = 5 and r range from 1 to n; then change n to 10, say; and make r range from, say, 5 to 9 … as you wish.

Duncan Williamson

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