A designer has a palette of fourteen colors to create a flag.
The flag must have four vertical stripes as pictured here.
Each stripe must be a different color. How many possible flags can be created?
So whenever we have n items, or in this case fourteen colors,
and we're selecting r items at a time, or in this case four colors for the stripes.
Our first question should be: Does the order matter?
If the order matters, then we're talking about a permutation.
If the order doesn't matter, then we're talking about a combination.
So looking at our flag here, let's just say,
one design looked like this, we want to ask ourselves,
if we change the order of these colors, would it produce a different flag?
And the answer is yes, and therefore, ordered does matter,
and therefore to solve this problem, we want to use a permutation.
So again because we have fourteen colors,
and we're selecting four colors at a time,
the total number of flags would be equal to
fourteen permute four. So if we have fourteen
permute four, looking at our formula here,
this means n equals fourteen, and r equals four.
So to evaluate this, we would have fourteen factorial
divided by fourteen minus four factorial.
So simplifying, we have fourteen factorial
divided by ten factorial. So to simplify this we'll begin to
expand fourteen factorial, so we'd have fourteen
times thirteen, times twelve, times eleven.
Now the remaining factors would be ten factorial,
so we'll stop here because, notice how we have ten factorial in the denominator.
So we have ten factorial here, and divided by ten factorial,
of course this simplifies to one. So we're left with fourteen
times thirteen, times twelve, times eleven
which is equal to twenty-four thousand twenty-four.
So there are twenty-four thousand twenty-four
possible flags. I also wanted to show how we could have evaluated divisional
permutation
on the calculator, using this notation here.
So back to the calculator, for fourteen permute
four, we'd first enter fourteen, which is n. Then press math,
right arrow to probability. Notice
option two is for permutations, so we'd press two.
And now we enter r, which is four,
and enter. But we should also be able to evaluate a permutation by hand
as shown here. Let's also talk about a second way
to answer this question using the counting principles. If we viewed selecting the
four colors for the stripes
as four separate events, we could use the counting principle
to find the same result. So to model the counting principle,
we would use four vertical bars to represent the four events.
Let's call this first event one C sub one,
for selecting the first color followed by C sub two,
C sub three, and C sub four. If we're going to determine the number of ways
each of these events can occur, the product of the number of ways
will give us a total number of ways, or the total possible flags.
So because there are fourteen colors to start with,
there are fourteen ways to select the first color. So we'd have fourteen here,
times, now the number of ways to select the second color will only be thirteen
because only one color has already been used here,
and there's no repetition allowed, so, times thirteen.
To select the third color, there would only be twelve ways, because two colors have already been
selected,
and eleven ways to select the fourth color. Notice how this product
is the same product we have here, using the permutation.
This would be twenty-four thousand twenty-four.
So there is more than one way to solve this problem.
I hope you found this helpful.