This article is written primarily for use with Business Students who deal with chance and randomness.
Factorials
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 | 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 (10^{11}) people ever lived SourceThere are also about 100 billion stars in the galaxy Source |
15! | 1.31×10^{12} | And about 500 billion galaxies in the Universe Source |
16! | 2.09×10^{13} | Wealth of world in dollars – $44 trillion dollars (that’s 4.4 x 10^{13}) Source |
17! | 3.56×10^{14} | And we’re still only at Factorial 17 |
18! | 6.40×10^{15} | How far is a light year? 9,460,536,207,068,016 metres (9.4 x 10^{15}) Source |
19! | 1.22×10^{17} | 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 10^{17} seconds. And we’re not up to factorial twenty yet! |
20! | 2.43×10^{18} | |
21! | 5.11×10^{19} | |
22! | 1.12×10^{21} | |
23! | 2.59×10^{22} | 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 × 10^{22} 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! | 6.20×10^{23} | |
25! | 1.55×10^{25} | In chemistry a mole is the number of grams of a substance equivalent to its molecular mass. For water, H_{2}O, 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 10^{23} 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 10^{25}. |
26! | 4.03×10^{26} | If you want to find the size of universe then there are many different theories. Figures vary, but about 10^{27} metres across may sound reasonable. What is certain is that no-one will ever know for sure. |
27! | 1.09×10^{28} | |
28! | 3.05×10^{29} | |
29! | 8.84×10^{30} | |
30! | 2.65×10^{32} | |
31! | 8.22×10^{33} | |
32! | 2.63×10^{35} | |
33! | 8.68×10^{36} | |
34! | 2.95×10^{38} | |
35! | 1.03×10^{40} | |
36! | 3.72×10^{41} | |
37! | 1.38×10^{43} | |
38! | 5.23×10^{44} | |
39! | 2.04×10^{46} | |
40! | 8.16×10^{47} | |
41! | 3.35×10^{49} | If you ask how many atoms there are in the earth you get to about 1.3 × 10^{50} Source |
42! | 1.41×10^{51} | We’re only at factorial 42. |
43! | 6.04×10^{52} | |
44! | 2.66×10^{54} | |
45! | 1.20×10^{56} | |
46! | 5.50×10^{57} | Atoms in the sun? about 1.19 × 10^{57} Source |
47! | 2.59×10^{59} | |
48! | 1.24×10^{61} | |
49! | 6.08×10^{62} | |
50! | 3.04×10^{64} | |
51! | 1.55×10^{66} | |
52! | 8.07×10^{67} | |
53! | 4.27×10^{69} | Atoms in our galaxy? About 1.2 x 10^{68} Source |
54! | 2.31×10^{61} | |
55! | 1.27×10^{73} | |
56! | 7.11×10^{74} | |
57! | 4.05×10^{76} | |
58! | 2.35×10^{78} | |
59! | 1.39×10^{80} | 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 10^{80} seems to be a reasonable guess. No one’s ever going to be able to count them, anyway. |
60! | 8.32×10^{81} | |
61! | 5.08×10^{83} | |
62! | 3.15×10^{85} | |
63! | 1.98×10^{87} | |
64! | 1.27×10^{89} | |
65! | 8.25×10^{90} | |
66! | 5.44×10^{92} | |
67! | 3.65×10^{94} | |
68! | 2.48×10^{96} | |
69! | 1.71×10^{98} | At 9.99 x 10^{99} most calculators give up. |
70! | 1.20×10^{100} | A Googol is defined at 10^{100}, 1 with a hundred zeros after it. Source |
80! | 7.16×10^{118} | From now on numbers go up in tens |
90! | 1.49×10^{138} | |
100! | 9.33×10^{157} | |
110! | 1.59×10^{178} | |
120! | 6.69×10^{198} | |
130! | 6.47×10^{219} | |
140! | 1.35×10^{241} | |
150! | 5.71×10^{262} | |
160! | 4.71×10^{284} | |
170! | 7.26×10^{306} | At this point Excel 2003 gave up. |
About 10^{12,978,189}
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 2^{43,112,609} -1 precisely | |
10^{googol}
10^{10100} |
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.
very good material… well done !
By: pythonisms on Wednesday December 31 2008
at 12:37:17 UTC