Current time:0:00Total duration:2:58

0 energy points

Studying for a test? Prepare with this lesson on AIME.

See lesson

# 2003 AIME II problem 3

Video transcript

Define a "good word" as
a sequence of letters that consists only of
the letters A, B, and C. Some of these letters may
not appear in the sequence. And in which A is never
immediately followed by B, B is never immediately
followed by C, and C is never
immediately followed by A. How many seven letter
"good words" are there? So let's just think
about this a little bit. So there's letters with
just A's, B's, and C's. And then it could be all
A's, all B's, all C's because some letters
might not appear. And A is never
immediately followed by B. So A can only be followed
by another A or another C. B is never immediately
followed by C, which means that B could only be
followed by an A or another B. And C is never
immediately followed by A. So C could only be followed
by another C or a B. So how many seven letter
"good words" are there? So let's just think
about the places. We have seven letters, so
1, 2, 3, 4, 5, 6, 7 letters. Now, there's no constraints
on this first letter since it's not
following anything. So it could be an A
B, or a C. So there's three possibilities
for this first letter. Now, there's three possibilities
for this first letter. But no matter what letter this
is, how many possibilities are there for this second
letter over here? Well, if this was an
A, the second letter could only be an A
or a C because it can't be followed by
a B. If this was a B, the second letter could
only be a B or an A because it can't be followed
by a C. If this was a C, the second letter could
only be a B, or a C. So no matter what letter
this first letter is, the second letter can only
have two possibilities. There could only be
two possibilities. Another way to
think about it is, there's one letter, no
matter what letter this is, there's one letter that's
being ruled out here. So it could only be
two possibilities. Well, the same thing is here. We're going to stick
some letter here. And no matter what letter
there is over here, it's going to rule out
one possibility over here. So we're going to have
only two possible letters that we can put here, no
matter what letter is there. And use the same logic, only
two possibilities there, only two possibilities there,
only two possibilities there, and then only two
possibilities there. So how many total
possibilities do we have? Well, 3 times 2 times 2
times 2 times 2 times 2. This is 1, 2, 3, 4, 5, 6 2's. So this is equal to 3
times 2 to the sixth power, which is 3 times 2 to
the six is 32 times 2 is 64, which is equal to
180 plus 12 is equal to 192. There's 192 possible
good seven letter "good words," where good
words is defined above.