Say you have 12 books, but room to display only 7. I will open this to everyone - how many differnt arrangements are there if the order of the books matters (permutation) and how many if order is of no consequence (combination)?
lajnklh
If the order does matter than the number of permutations is 12*11*10*9*8*7*6= P(12,7) = 3991680
if the order does not matter (combinations)
it would be 12!/(7!5!) = C(12,7) = 792
both P(n,r) and C(n,r) can be found on a standard scientific calculator
usually they are labeled nPr and nCr
to do C(12,7)
enter
12
2nd F
nCr
7
=
you should get 792
To solve this problem, we can use the concepts of permutations and combinations.
First, let's calculate the number of different arrangements (permutations) when the order of the books matters.
Permutations:
The number of permutations can be calculated using the formula for permutation of r objects from n total objects, which is denoted as P(n,r). The formula for permutations is:
P(n,r) = n! / (n - r)!
In this case, we have 12 books and we want to choose 7 to display. So, n = 12 and r = 7.
Using the formula, we can calculate the number of permutations:
P(12,7) = 12! / (12 - 7)!
= 12! / 5!
= (12 * 11 * 10 * 9 * 8 * 7!) / 5!
= (12 * 11 * 10 * 9 * 8)
= 66,5280
There are 66,5280 different arrangements (permutations) when the order of the books matters.
Now, let's calculate the number of different arrangements (combinations) when the order of the books doesn't matter.
Combinations:
The number of combinations can be calculated using the formula for combination of r objects from n total objects, which is denoted as C(n,r). The formula for combinations is:
C(n,r) = n! / (r! * (n - r)!)
In this case, we again have 12 books and we want to choose 7 to display. So, n = 12 and r = 7.
Using the formula, we can calculate the number of combinations:
C(12,7) = 12! / (7! * (12 - 7)!)
= 12! / (7! * 5!)
= (12 * 11 * 10 * 9 * 8 * 7!) / (7! * 5!)
= (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
= 792
There are 792 different arrangements (combinations) when the order of the books doesn't matter.