Mathematics

[druiaghtagh]

If you take 20 letters of the alphabet, what is the maths for calculating all possible combinations using all 20 leters each time please?

[Gef]

My guess would be 20! (20 factorial) which is 20x19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1 which is 2,432,902,000,000,000,000 or 2 million 4 hundred and 32 thousand 9 hundred and 2 million million.

[lisaj]

Gef's guess is right. To begin with you have 20 choices, then you have 19 remaing choices, then 18 and so on.

[sft42]

A lot then?

[mikewall01]

The above answers are incorrect.
A combination is a particular set of symbols independent of their ordering or arrangement. So the answer to your question is one
I believe that what you might want to know is how many permutations there are. Permutation is an ordering; a rearrangement, of the symbols.
There are 2432902008176640000 permutations of 20 symbols.
You asked also what is the maths. I will post a web site that will show you how to make the calculation

[mikewall01]

There are two links below that show how to calculate both combinations and permutations
http://www.wcrl.ars.usda.gov/cec/h.htm#javscrt
and this one;
http://www.ciphersbyritter.com/JAVASCRP/PERMCOMB.H
TM
you will see that they both calculate the perutations as 2,432,902,008,176,640,000 which is quite a lot of small change.

[Squirrel]

Well, mikewall01, if you wanted to be very accurate indeed you could call the above answers incorrect, but the percentage difference falls into the millionths of 1%. As you know, ⁿP_r is the symbol for permutations, where n is the big number and r is the small one. The way I think of it is 'The number of ways of choosing r objects out of a bag containing n objects, and then arranging those r objects in every possible order.' Hence an alternative way of calculating permutations is to consider the factorial of r (the number of ways of arranging r objects in any order), and then multiply it by ⁿC_r (the combinations of choosing r objects out of a bag containing n; for example, the number of possible ways that you could knock down 8 bowling pins and leave 2 standing would be ¹⁰C₂ on a calculator, or [10!/(10-2)!x2!] if you wanted to do it on paper.) What this comes down to is that if the permutation calculation is rearranged for the alphabet question, it gives:

ⁿC_r x r! = ⁿP_r

²⁰C₂₀ = (20! / (20!)²) x 20! = 1 / 20! x 20!

The above will obviously be equal to one, so going back to the first line of the calculation:

1 x 20! = ²⁰P₂₀

(Exceeding 2000 characters now')

[Squirrel]

I hope I have explained this OK, but to summarise so you will remember in future:

Combinations. The number of ways of picking r balls out of a bag containing n, then holding them in your hand, so they are in no particular order. (Example: number of ways of selecting a football team out of a group of 30)

Permutations. The number of ways of picking r balls out of a bag containing n, then arranging the balls you picked in every possible order. (Example: number of ways of putting 3 eggs in 5 egg cups.)

P.S. Looks like I left the italic on in the above post - oops.

[ukkid76]

Why would you want to know??????????

[Gef]

mikewall01, I know what druiaghtagh was asking! and by the way if my answer was slighty incorrect (I can't be bothered checking) it's because I was using the very basic in-built calculator on my i-Mac and not a scientific one :-)