Question 1116689: A computer user has downloaded 25 songs using an online file-sharing program and wants to create a CD-R with ten songs to use in his portable CD player. If the order that the songs are placed on the CD-R is important to him, how many different CD-Rs could he make from the 25 songs available to him?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! permutation formula is used for this.
that is p(n,x) = n! / (n-x)!
n is the number you are choosing from.
x is the number you are choosing.
in your problem:
n = 25 an x = 10
you get p(25,10) = 25! / (25 - 10)!.
this becomes p(25,10) = 25! / 15!
this is the same as p(25,10) = (25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15!) / 15!.
the 15! in the numerator and the denominator cancel out and you are left with:
p(25,10) = (25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16).
use your calculator to get p(25,10) = 1.186167629 * 10 ^ 13.
if your calculator has a permutation formula routine, use that and it will tell you the same thing.
i have a TI-84 Plus.
i enter 25 -> -> -> 2 10 and it give me the same answer.
that would be:
hit the math key, then 3 arrows to the right to get into PRB mode, then 2 to get into nPr mode, then 10.
that's the same thing as 25P10 which is the same as p(25,10).
other scientific calculators can do the same thing, but not necessarily with the same key presses.
if order matters, use the permutation formula.
if order doesn't matter, use the combination formula.
the difference in the formulas is a division by x!.
permutation formula is p(n,x) = n! / (n-x)!
combination formula is c(n,x) = n! / (x! * (n-x)!)
that division by x! is what removes the ordering requirement.
here's a simple example to show you what that mean.
assume n = 3 and x = 2
assume you are selecting from a set that contains a,b,c
p(n,x) gets you 3! / 1! = 3 * 2 = 6
c(n,x) gets you 3! / (2! * 1!) = (3 * 2) / 2 = 3
the ordered sets are:
a,b
a,c
b,c
b,a
c,a
c,b
the unordered sets are:
a,b
a,c
b,c
with the ordered sets:
a,b and b,a are different sets because order is important to define the set.
with the unordered sets:
a,b and b,a are the the same set because order is not important to define the same set.
since a,b and b,a are the same elements, only in a different order, then you count only a,b and not b,a.
consider the set a,b.
how many different ways can you arrange the elements.
that would be x!, where x = 2.
2! = 2*1 = 2
the set a,b can be ordered in 2! ways.
you would get a,b and b,a.
to remove the order, you divide by 2! to get 1 possible way when order doesn't matter.
that's what the c(n,x) formula does.
it's the p(n,x) formula with an additional division by x!.
a,b and b,a becomes simply a,b.
some formulas use different variables, such as c(n,r) or nPr.
same formula only they use r instead of x.
in your problem p(25,10) = 1.186167629 * 10^13.
c(25,10) = 3268760
take 1.186167629 * 10^13 and divide it by 10! and you get 3268760.
10! is equal to 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
here's a reference:
https://www.mathsisfun.com/combinatorics/combinations-permutations.html
note that the combination and permutation formulas used here assume that the same element cannot be used more than once in each set.
you would not have a cd-r that has 10 songs on it that are exactly the same.
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