SAT Prep: Test 2 Section 6 Part 6

Problems 17-18 on page 476

Welcome back.

So I did this ahead of time, so
as to not waste your time

drawing it.

The question says, a merchant
sells three types of clocks

that chime as indicated
by the check marks

in the table above.

What is the total number of
chimes of the inventory of

clocks in the 90-minute
period?

And we're talking about
7:15 to 8:45.

So let's just make sure we
understand this chart.

So we have clocks of
type a, b, and c.

This is the number of
each of those clocks

that the store has.

Now these n times on the nth
hour, that means when it turns

7 o'clock, it's going to
chime seven times.

When it turns 8 o'clock, it's
going to chime eight times, et

cetera, et cetera.

This is once on the hour,
so right at the hour

does just one chime.

And then these do once
on the half hour.

So let's see, we have
clock a times 10.

And how many times is each of
the clock a's going to chime

between this period?

Well, it does n times
on the nth hour.

So between 7:15 and 8:45,
there's only one hour that

happens, which is 8 o'clock.

And at 8 o'clock it's going
to chime n times.

It's going to chime eight
chimes at 8 o'clock.

And it also does one
on the half hour.

So what half hours are there?

Well there's 7:30 and there's
8:30 that pass up.

So there are two half hours.

It's going to chime at 7:30
and 8:30, each of these.

So then plus 2, once at 7:30
and once at 8:30, times 10,

because there's ten clocks.

Let's see, b times 5 clocks,
n times on the nth hour.

Well, once again that's eight
chimes at 8:00, and it doesn't

do anything else.

So times 5.

And then finally c times
three clocks.

It doesn't do this n
times the nth hour.

It does once on the hour.

It's going to do it
once at 8 o'clock.

And then once on the half
hour, so plus 1 at

7:30 plus 1 at 8:30.

So that's 3 times
three clocks.

So what's this?

It's 8 plus 2.

Eight chimes plus two chimes.

So that's ten chimes per
clock times 10 is

equal to a 100 chimes.

Eight chimes per clock
times five clocks

is equal to 40 chimes.

And then three chimes per clock
times three clocks is

equal to 9.

You add all this up together,
and the whole store in this

time period, if I haven't made
a mistake, will chime 149 or

there'll be 149 chimes of the
inventory of clocks in this

90-minute period.

Next question.

So they draw a bunch
of random things.

I'll just replace them with
numbers because that's easier

for me to think, or letters
a, b, c, d, and e.

I mean those pictures
are useless.

They say, if the five cards
shown above are placed in a

row so that the shaded box, but
I'll call that c, so that

c is never at either end, how
many different arrangements

are possible?

Well, this is how I
think about it.

c is the most restricted.

Let's say that we're placing
them in order.

Let's say these are the spots.

They're in a row.

1, 2, 3, 4, 5.

And when we come along,
we have c.

We know that c can't be
placed here, and it

can't be placed here.

So it can only be placed in
one of these three spots.

So there's three possible
situations

where c can be placed.

Let's say you place c in one
of these three spots.

Now it's our turn to place a.

How many spots are
left to place a?

Well, one of these three is
going to be taken by c.

But then a can be in any
of the other four.

So then there's four
possibilities for a.

And then once you place a, then
for b, well two of the

spots are going to be
taken up in b, so

there's three spots left.

So there's three possibilities
for b.

And then there's only
two openings for d.

And then there's
only one for e.

So if you multiply this out,
you get 3 times 4 is 12.

12 times 3 is 36.

36 times 2 is 72.

So there are 72 possible
arrangements where the middle

card or that grey card is never
placed at either end.

That's how I always think about
these because you just

think about placing them-- well
this first one can only

go in three-- well
you get the idea.

Hopefully you get the idea.

Well, that's this section,
so I will see

you in the next section.

Have fun.