This lesson we'll look
at several examples of permutations.
A permutation is an arrangement when order matters.
A permutation is an ordered list with no repetition.
A combination is an arrangement when order does not matter.
A combination is for grouping with no repetition.
In our first example, there are five children
that are taking turns at using the swing on a playground.
How many ways are there
in which they can line up to use the swing
when everyone gets exactly one turn?
So the order of the lineup matters
and there's no repetition,
and therefore, this is an example of a permutation.
Since there are five children
and they all want to use the swing,
the number of ways they can line up is five permute five,
which can be written using this notation,
five permute five, or this notation, five permute five.
n permute five is equal to n factorial
divided by n minus r factorial.
In our case, both n and r are equal to five,
and therefore, this is equal to five factorial
divided by five minus five factorial,
which is equal to five factorial divided by zero factorial.
Zero factorial is equal to one,
and therefore, we have five factorial,
which is equal to five times four
times three times two times one,
which is equal to 120.
There are 120 ways
in which the five children can line up to use the swing.
Let's just verify this on the calculator.
There's a couple of ways of doing this,
you can type in nPr, or go to functions,
click nPr, go back to the main screen,
and enter five comma five, Enter, which verifies our result.
Or we can enter five factorial by clicking five,
going to functions and clicking the factorial
and then Enter.
The next example,
Tony has 14 books on this summer reading list.
He wants to choose four books to read in order
from first to last.
In how many ways can he do this?
So once again, we do have a permutation
because the order matters, from first to last.
And we are assuming he will not read the same book twice.
And therefore, the number of ways
he can choose four books to read from first to last
from the 14 books is 14 permute four.
Again, normally, you would not write both notations,
just pick one and use it.
In this case, we have n is equal to 14
and r is equal to four,
which means 14 permute four is equal to 14 factorial
divided by 14 minus four factorial.
Which is equal to 14 factorial divided by 10 factorial,
let's go and simplify this by hand,
and then verify it on the calculator.
Let's expand 14 factorial,
14 factorial is equal to 14 times 13 times 12
times 11 times, the remaining factors would be 10 factorial,
so let's write 10 factorial.
Then we have divided by 10 factorial,
10 factorial divided by itself simplifies to one,
and therefore, we're left with 14
times 13 times 12 times 11.
Let's go ahead and verify this on the calculator.
14 times 13 times 12 times 11 is equal to 24,024.
Let's also verify that 14 permute four
gives us the same result.
So we can either type in nPr or go to functions,
click nPr, go back to the main number keypad,
enter 14 comma four,
and notice how we do get the same result.
24,024 ways
Tony can pick his books.
Next, the student council has 11 students,
they need to choose a President, Vice President,
Secretary, and Treasurer for Student Council.
In how many ways can the four different positions
be appointed from the 11 students?
Once again, the order here does matter
because the order indicates which position
the student will take on the student council.
And of course, we don't have repetition,
because the student cannot hold two offices.
And therefore the number of ways
the four different positions can be appointed
from 11 students is 11 permute four.
Where here n is equal to 11 and r is equal to 4.
11 permute four is equal to 11 factorial
divided by 11 minus four factorial.
Simplifying we have 11 factorial
divided by seven factorial.
Now let's go ahead and expand 11 factorial
until we get to seven factorial.
11 factorial is equal to 11 times 10
times nine times eight times the remaining factors
would be seven factorial, divided by seven factorial.
Seven factorial divided by itself simplifies to one.
We're left with 11 times 10 times nine times eight.
11 times 10 times nine times eight is equal to 7,920.
And 11 permute four should give us the same result.
Click functions, click nPr, type in 11 comma four,
and we do get the same result.
There are 7,920 ways
the four positions can be appointed from the 11 students.
The last example, Zach is creating a seven-digit passcode
using the digits zero through nine.
How many seven-digit passcodes are possible
if the numbers cannot be repeated?
So once again, we don't have repetition
and the order does matter.
And therefore, we can answer this question
using a permutation.
Zero through nine gives us 10 digits.
And because Zach is selecting seven-digit passcodes,
the number of possible passcodes under these conditions
would be 10 permute seven.
Where n is equal to 10 and r is equal to seven,
which gives us 10 factorial
divided by 10 minus seven factorial.
Simplifying, we have 10 factorial
divided by three factorial.
Let's now expand 10 factorial
until we reach three factorial,
so 10 times nine times eight times seven
times six times five times four,
the remaining factors are equal to three factorial
divided by three factorial.
Simplifying,
and now we find the product.
We have 604,800 possible passcodes.
Let's also verify we get the same result
for 10 permute seven,
and of course, we do.
I hope you found this helpful.