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# Ways to arrangeĀ colors

Thinking about how many ways you can pick four colors from a group of 6. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

In one game, a code made using
different colors is created by one player, the codemaker,
and the other player, the codebreaker, tries to
guess the code. The codemaker gives hints about
whether the colors are correct and in the
right position. All right. The possible colors are blue--
let me underline these in the actual colors-- blue, yellow,
white, red, orange and green. Green is already written in
green, but I'll underline it in green again. And green. How many 4-color codes
can be made if the colors cannot be repeated? To some degree, this whole
paragraph in the beginning doesn't even matter. If we're just choosing from--
let's see, we're choosing from-- how many colors
are there? There's 1, 2, 3, 4, 5, 6 colors,
and we're going to pick 4 of them. How many 4-color codes
can be made if the colors cannot be repeated? And since these are codes, we're
going to assume that blue, red, yellow and green,
that this-- that that is different than green, red,
yellow and blue. We're going to assume that these
are not the same code. Even though we've picked the
same 4 colors, we're going to assume that these are 2
different codes, and that makes sense because we're
dealing with codes. So these are different codes. So this would count as 2
different codes right here, even though we've picked
the same actual colors. The same 4 colors,
we've picked them in different orders. Now, with that out of the way,
let's think about how many different ways we can
pick 4 colors. So let's say we have
4 slots here. 1 slot, 2 slot, 3 slot
and 4 slots. And at first, we care only
about, how many ways can we pick a color for that slot right
there, that first slot? We haven't picked
any colors yet. Well, we have 6 possible colors,
1, 2, 3, 4, 5, 6. So there's going to be 6
different possibilities for this slot right there. So let's put a 6 right there. Now, they told us that the
colors cannot be repeated, so whatever color is in this slot,
we're going to take it out of the possible colors. So now that we've taken that
color out, how many possibilities are when we go
to this slot, when we go to the next slot? How many possibilities
when we go to the next slot right here? Well, we took 1 of the 6 out for
the first slot, so there's only 5 possibilities here. And by the same logic when we
go to the third slot, we've used up 2 of the slots-- 2 of
the colors already, so there would only 4 possible
colors left. And then for the last slot, we
would've used up 3 of the colors, so there's only
3 possibilities left. So if we think about all of the
possibilities, all of the permutations-- and permutations
are when you think about all the
possibilities and you do care about order; where you say that
this is different than this-- this is a different
permutation than this. So all of the different
permutations here, when you pick 4 colors out of a possible
of 6 colors, it's going to be 6 possibilities for
the first 1, times 5 for the second bucket, times 4 for
the third or the third bucket of the third position,
times 3. So 6 times 5 is 30, times
4 is times 3. So 30 times 12. So this is 30 times 12, which is
equal to their 360 possible 4-color codes.