D. I. Y. Investing
How to manage your financial affairs
Permutations – or how to win the Lotto
On Friday 24 October 2014, the Euromillions jackpot of € 190 million was won by a single ticket bought by a resident of the city of Castelo Branco in Portugal. The winning numbers were 3, 9, 20, 30, 42 plus two stars 1 and 6. The odds of winning a jackpot like this are virtually zero but someone did. The winnings are taxed at 20% and if not claimed within three months are forfeited.
The odds of winning the main Euromillions prize are 116,531,800 to 1.
On Saturday 27 September 2014, a former Bosasa employee won the S.A. Lotto main prize of R15 million and promptly resigned. So what’s to stop you and me from choosing our six numbers and winning the LOTTO on a Wednesday or a Saturday evening and retiring to Cape Town, Port Alfred, Springbok or Maputo? And why don’t more people win the LOTTO more often? There is nothing to stop us winning; all we have to do is to choose the correct six numbers out of forty nine! In order to win the lottery we only have to find EXACTLY the right combination of six numbers that will win us the millions we dream of.
If we have to choose 6 numbers and there are 49 numbers to choose from on the lottery ticket, there are 13,983,816 ways to do that. That means that the lucky Bosasa winner overcame odds of nearly fourteen million to one to do just that.
That is why it is so difficult to win the lottery. So the permutations for winning the Lotto are huge numbers [13,983,816 from 6 choices out of 49 numbers] that develop out of much smaller numbers. Let’s shrink the numbers now and get an understanding of permutations to see exactly what it is that we are trying to achieve.
Permutations – a permutation is defined as being an ordered arrangement or grouping of a set of numbers, items etc. and any one of a range of possible groupings, according to the Concise Oxford Dictionary.
Let’s take a simple example of what a permutation really means:
Imagine that you are can choose two dishes out of a three dish menu: a starter, a main and a desert. We want to know how many ways in which we can select our two dishes from the three on offer:
1 Main, Desert
2 Main, Starter
3 Desert, Main
4 Desert, Starter
5 Starter, Main
6 Starter, Desert
We find a total of six permutations of our three dishes if we want to choose them in groups of two. Now imagine that we have been given greater choice: now we have four dishes to choose from. We want to know the number of ways in which we can choose two dishes from four: Main, Desert, Starter and Cheese and biscuits: Here we are:
1 Main, Desert
2 Main, Starter
3 Main, Cheese and biscuits
4 Desert, Main
5 Desert, Starter
6 Desert, Cheese and biscuits
7 Starter, Main
8 Starter, Desert
9 Starter, Cheese and biscuits
10 Cheese and biscuits, Main
11 Cheese and biscuits, Desert
12 Cheese and biscuits, Starter
a total of 12 permutations or ways in which we can put together two choices from the four available. Now add Coffee to the menu. How many ways are there now in which you can choose two offerings from the five on offer?
Did you get 20 different ways? Good! The method used above is laborious and we need an elegant solution. For example, if you are about to choose your courses at university and you have fifteen subjects to choose any three from and you are interested to find out how many ways in which you might choose your three subjects, you wouldn’t want to make a list of them like we’ve been doing so far. There are after all 2 730 permutations of 3 from 15, 32 760 when we are permutating 4 from 15, and the list goes on, and on ad infinitum. And now for some elegant confusion;
Mathematicians to the Rescue! As ever, the world’s mathematicians have done all the necessary work for us and they have developed a relatively simple formula for us to work with as we seek answers to these permutation problems. Permutation Formula: n!/(n-r)!) where n is the group size, r is the number of items to be selected, without replacement from the group; n! is read as n factorial and where, for example, n is 5 it means n! = 5 x 4 x 3 x 2 x 1 (n – r)! means the factorial of the result of having subtracted r from n.
Reworking our examples we can use this formula now, starting with the smallest and easiest numbers:
3 from 15 subjects: 15!/(3!*(15-3)!) = 455 – 6 from 45 lottery numbers: 45!/(6!*(45-6)!) = 8,145,060 – 6 from 49 lottery numbers: 49!/(6!*(49-6)!) = 13,983,816 etc.
If I have lost you by now, and I am sure that I have, I am going to show you a shortcut. You have to choose 6 numbers out of 49 and once selected a number cannot be drawn again. This is how you do it 49 x 48 x 47 x 46 x 45 x 44 divided by 6 x 5 x 4 x 3 x 2 x 1 = 10,068,347,520 / 720 =13,983,816. Easy, no?
Let’s compare the Vodacom Millionaires competition to the Lotto. You have to select 9 letters out of 26 letters and to compound matters letters are not removed from selection once selected, but remain in the draw. So the number of permutations is 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 = 1,133,836,704,000 – you have a better chance of becoming a Lotto or a Euro millions millionaire than a Vodacom millionaire.
Moral of the story! The odds are stacked against you. You would be better off investing your money in a sensible investment – and that is why mathematics is so important – it gives you the knowledge to work out the odds.
On 28 May 2013 the winning letters in the Vodacom competition were M W A M L K V D X, the M was drawn twice, ouch!On Friday 24 October 2014, the Euromillions jackpot of € 190 million was won by a single ticket bought by a resident of the city of Castelo Branco in Portugal. The winning numbers were 3, 9, 20, 30, 42 plus two stars 1 and 6. The odds of winning a jackpot like this are virtually…
Here is exact numbers of lines for the set of numbers.
2 NUMBERS PERMING (P2)
47= 1081 (LINES)
3 NUMBERS PERMUTATION (P3)
47= 16,215 (LINES)
4 NUMBERS PERMUTATION (P4)
2 thoughts on “PERMUTATIONS CHART”
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NB: Your SUB begins immediately your account is upgraded.Here is exact numbers of lines for the set of numbers. 2 NUMBERS PERMING (P2) 2=1 (LINES) 3=3 (LINES) 4=6 (LINES) 5=10 (LINES) 6=15 (LINES) 7=21 (LINES) 8=28 (LINES) 9=36 (LINES) 10=45 (LINES) 11=55 (LINES) 12=66 (LINES) 13=78 (LINES) 14=91 (LINES) 15=105 (LINES) 16=120 (LINES) 17=136 (LINES) 18=153 (LINES) 19=171 (LINES) 20=190 … ]]>