Factorials

This article is written primarily for use with Business Students who deal with chance and randomness.

Ever wondered what that key on your calculator with x! on it did? If you chanced to look it up in the manual you’d have found it was the factorial key, and if you’d pressed it something less than comprehensible may have turned up.

A factorial is the product of all integers from the number you want the factorial for down to one, subtracting one each time.

x! = x ×  (x – 1) × (x – 2) × … × 3 ×  2 × 1

The ellipsis (…) in the middle means that you just continue with the same maths until you meet up with the other end.

So:

4! = 4 × 3 × 2 × 1

4! = 24

And:

7! = 7 × (7 – 1) × (7 – 2) × … × 3 × 2 × 1

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 5040

Now that’s a bit of a difference – possibly more than you’d expect. By the time you get to 10! you’re in the millions. It seems such a simple, innocent sum, really.

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

10! = 3,628,800

We’ll look at why all this is necessary after you’ve had a look at the table below.

Factorial and other values

Factorial	Number	Setting the scene with examples
0!	1	The factorial of zero is 1 Source
1!	1	The factorial of 1 is also 1
2!	2	2! = 2 × 1 = 2
3!	6	3! = 3 × 2 × 1 = 6
4!	34	4! = 4 × 3 × 2 × 1 = 24
5!	120	5! = 5 × 4 × 3 × 2 × 1 = 120
6!	720	6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
7!	5,040	and so on
8!	40,320
9!	362,880
10!	3,628,800
11!	39,916,800	Chance getting six numbers in the UK National Lottery: 1 in 13,983,816 There were 49,138,831 people in England at the 2001 census. Source There were 60,975,000 people in Britain in mid-2007 Source Chance getting five numbers and two stars in the Euro Millions Lottery: 1 in 76,275,360
12!	479,001,600	Population of the European Union on 1st January 2009 (Estimate) 499.7 million Source
13!	6,227,020,800	The 2008 estimate for the world population is about 6.7 billion. Source
14!	87,178,291,200	About 100 billion (1011) people ever lived SourceThere are also about 100 billion stars in the galaxy Source
15!	1.31×1012	And about 500 billion galaxies in the Universe Source
16!	2.09×1013	Wealth of world in dollars – $44 trillion dollars (that’s 4.4 x 1013) Source
17!	3.56×1014	And we’re still only at Factorial 17
18!	6.40×1015	How far is a light year? 9,460,536,207,068,016 metres (9.4 x 1015) Source
19!	1.22×1017	The age of the Universe, i.e since the Big Bang, is between 13.6 and 13.8 billion years (Source) A year, currently is about 365.25 days and there are 24 × 60 × 60 seconds in a day. So if we multiply all these together we get the age of the Universe in seconds, a mere 4.3 x 1017 seconds. And we’re not up to factorial twenty yet!
20!	2.43×1018
21!	5.11×1019
22!	1.12×1021
23!	2.59×1022	So how many stars are there in the Universe? You can only estimate, but if we take ours as a sort of normal, run of the mill galaxy, with 100 billion stars, and there are 500 billion galaxies, what we have to do is multiply them together. Then we get about 5 × 1022 stars. If we write this out we get 50,000,000,000,000,000,000,000 stars.  Quite big, but we’re still only at factorial 23.
24!	6.20×1023
25!	1.55×1025	In chemistry a mole is the number of grams of a substance equivalent to its molecular mass. For water, H2O, this is 18g, so there are 1000/18 = 55.56 moles in a litre of water. Each mole of any substance contains 6.02 x 1023 molecules (Avogadro’s Number Source), so to find the number of molecules of water in a litre you multiply them together and get 3.34 x 1025.
26!	4.03×1026	If you want to find the size of universe then there are many different theories. Figures vary, but about 1027 metres across may sound reasonable. What is certain is that no-one will ever know for sure.
27!	1.09×1028
28!	3.05×1029
29!	8.84×1030
30!	2.65×1032
31!	8.22×1033
32!	2.63×1035
33!	8.68×1036
34!	2.95×1038
35!	1.03×1040
36!	3.72×1041
37!	1.38×1043
38!	5.23×1044
39!	2.04×1046
40!	8.16×1047
41!	3.35×1049	If you ask how many atoms there are in the earth you get to about 1.3 × 1050 Source
42!	1.41×1051	We’re only at factorial 42.
43!	6.04×1052
44!	2.66×1054
45!	1.20×1056
46!	5.50×1057	Atoms in the sun? about 1.19 × 1057 Source
47!	2.59×1059
48!	1.24×1061
49!	6.08×1062
50!	3.04×1064
51!	1.55×1066
52!	8.07×1067
53!	4.27×1069	Atoms in our galaxy? About 1.2 x 1068 Source
54!	2.31×1061
55!	1.27×1073
56!	7.11×1074
57!	4.05×1076
58!	2.35×1078
59!	1.39×1080	If we’re talking about atoms in universe then a simple answer might be ‘a lot’. How you define the universe is central to the argument, but about 1080 seems to be a reasonable guess. No one’s ever going to be able to count them, anyway.
60!	8.32×1081
61!	5.08×1083
62!	3.15×1085
63!	1.98×1087
64!	1.27×1089
65!	8.25×1090
66!	5.44×1092
67!	3.65×1094
68!	2.48×1096
69!	1.71×1098	At 9.99 x 1099 most calculators give up.
70!	1.20×10100	A Googol is defined at 10100, 1 with a hundred zeros after it. Source
80!	7.16×10118	From now on numbers go up in tens
90!	1.49×10138
100!	9.33×10157
110!	1.59×10178
120!	6.69×10198
130!	6.47×10219
140!	1.35×10241
150!	5.71×10262
160!	4.71×10284
170!	7.26×10306	At this point Excel 2003 gave up.
	About 1012,978,189  This is a number with close on 13 million digits	A prime number can only be exactly divided by itself and 1 – it has no other factors. Large prime numbers have significance in computer security and encryption. Mersenne primes have specific properties (Source) and the latest and largest was discovered in August 2008. The largest Mersenne Prime is 243,112,609 -1 precisely
	10googol  1010100	10 raised to the power of a googol is a Googloplex.To see it written out you’ll have to go to the source. If you were to attempt to write out a googolplex, you have to use more space than known universe occupies. But in number terms we’re not finished – not by a long chalk. An infinitely long chalk.
	∞  The symbol for infinity	Infinity is a very long way off! In fact, no one will ever get there.  The universe is finite. Numbers go on for ever.

Why?

In searching out probabilities for events to happen you must be able to know the likelihood of events happening, even though those events coud happen in a random manner. Suppose that you are in charge of maintenance for a call centre and need to predict the likelihood of failure of PCs. A failure of a PC means lack of productivity, so you keep a number of spare PCs in stock. How many spares you need can be calculate, and this calculation involves factorials.

Chance

If you’ve ever played the UK National Lottery main game, you’ll know that you choose six numbers from 49. The number selection is made using a machine with identically sized, differently numbered balls. The first ball our can be selected from a set of 49, the second from a set of 48 (because one has already gone) the third from 47 and so on. In order to calculate the chance of getting all your six numbers in the first six balls to drop out is an application of combinatorics, the branch of pure mathematics that studies discrete objects. The full arguments can be found from this source.

A future post on this blog will deal with chance and choice.