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# counting

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In how many ways can I arrange \$3\$ different math books and \$4\$ different history books on my bookshelf, if at least three of the history books are next to each other?

Mar 25, 2024

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+128794
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If three of the history books are together

They can occupy  positions 123 and  be arranged in 3! ways.....the other history book  can occupy positions 5, 6 or7

And the other three math books can  be arranged in 3! ways

So   3! * 3 * 3!   =  108 ways

They can occupy  positions  234 and  be  arranged in 3! ways.....the oher history book can occupy positions 6 or 7

And the three math books can  be arranged in 3! ways

So   3! * 2 * 3!  =  72 ways

And occupying positions  345 or 456   also  produce 2 ( 72)  arrangements   =144

And occupying  567   also produces another 108  arrangements

So  if three of the history books are together we have  2(108) +3(72)  = 432 arrangements

If all 4 of the history books  are together they can occupy positions

1 - 4

2 - 5

3 - 6

4 - 7

For each of these, they can  be arranged in 4! ways  and the other 3 math books can be arranged in 3! ways

So....if all are together, they can  be arranged in  4 * 4! * 3!   =  3456 ways

So....the total number of arrangements =  432 + 3456  =  3888 total arrangements

Mar 25, 2024