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Welcome back.

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

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drawing it.

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The question says, a merchant[br]sells three types of clocks

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that chime as indicated[br]by the check marks

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in the table above.

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What is the total number of[br]chimes of the inventory of

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clocks in the 90-minute[br]period?

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

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

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

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This is the number of[br]each of those clocks

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that the store has.

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

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

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

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cetera, et cetera.

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This is once on the hour,[br]so right at the hour

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does just one chime.

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And then these do once[br]on the half hour.

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

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

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between this period?

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Well, it does n times[br]on the nth hour.

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

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happens, which is 8 o'clock.

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

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

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And it also does one[br]on the half hour.

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So what half hours are there?

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

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So there are two half hours.

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

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

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because there's ten clocks.

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

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

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do anything else.

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So times 5.

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And then finally c times[br]three clocks.

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

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It does once on the hour.

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

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

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7:30 plus 1 at 8:30.

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So that's 3 times[br]three clocks.

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So what's this?

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It's 8 plus 2.

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Eight chimes plus two chimes.

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So that's ten chimes per[br]clock times 10 is

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equal to a 100 chimes.

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Eight chimes per clock[br]times five clocks

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is equal to 40 chimes.

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And then three chimes per clock[br]times three clocks is

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equal to 9.

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

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

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

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90-minute period.

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Next question.

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So they draw a bunch[br]of random things.

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

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

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I mean those pictures[br]are useless.

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

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

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c is never at either end, how[br]many different arrangements

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are possible?

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Well, this is how I[br]think about it.

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c is the most restricted.

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

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Let's say these are the spots.

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They're in a row.

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1, 2, 3, 4, 5.

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And when we come along,[br]we have c.

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

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can't be placed here.

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

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So there's three possible[br]situations

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where c can be placed.

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

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Now it's our turn to place a.

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How many spots are[br]left to place a?

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

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

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So then there's four[br]possibilities for a.

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

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spots are going to be[br]taken up in b, so

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there's three spots left.

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So there's three possibilities[br]for b.

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And then there's only[br]two openings for d.

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And then there's[br]only one for e.

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

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12 times 3 is 36.

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36 times 2 is 72.

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So there are 72 possible[br]arrangements where the middle

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card or that grey card is never[br]placed at either end.

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

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think about placing them-- well[br]this first one can only

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go in three-- well[br]you get the idea.

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Hopefully you get the idea.

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Well, that's this section,[br]so I will see

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you in the next section.

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Have fun.

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