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How many ways can you arrange 4 physics book and 5 math books so that the physics book are next to each other?

So I know the arrangements can be,

PPPPMMMMM
MPPPPMMMM
MMPPPPMMM
MMMPPPPMM
MMMMMPPPP

4! * 5! = 2880 ways
There are 4 ways to arrange the physics book and 5 ways to arrange the others so 4! * 5!. I'm not sure if that's the correct answer.

Kate
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3 Answers3

3

Doing it the long way, like you (but correctly !),

$\fbox{PPPP}MMMMM$

$M\fbox{PPPP}MMMM$

$MM\fbox{PPPP}MMM$

$MMM\fbox{PPPP}MM$

$MMMM\fbox{PPPP}M$

$MMMMM\fbox{PPPP}$

$6*4!*5!=17,280$

The short way is to realize that the block of $4$ Physics books have six places to start ($4$ gaps between $5$ Math books $+ 2$ ends) and then permute both groups, so (as before) $6*4!*5!$ ways

true blue anil
  • 41,295
2

Places to put the first Physics book: 6

Arrangements of Physics books: 4!

Arrangements of math books 5!

Total number of arrangements; $6\cdot 4! 5! = 17280$.

ncmathsadist
  • 49,383
  • I see! Thank you so much! – Kate Nov 01 '15 at 01:53
  • There are six places to place the first physics book since the number of math books that can be placed before the first physics book can range from $0$ to $5$, so your answer should be $6 \cdot 4!5!$. – N. F. Taussig Nov 01 '15 at 09:20
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Easier way to think of it, for me at least. Let "p" and "m" represent the physics and math books respectively. and let $P$ represent the physics books together. Then, there are 4! ways to arrange $P$ and 6! ways to arrange the remaining books like so:

  • Pmmmmm
  • mPmmmm
  • mmPmmm
  • mmmPmm
  • mmmmPm
  • mmmmmP

for a total of 6!4!=17280 ways. Same answer, just a slightly different way of thinking about it.

Marcus
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