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There are 10 numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1, 2, 3, 4, 5 have to be in that order, 6, 7, 8 have to be in that order, and 9, 10 have to be in that order, how many ways are there to arrange the ten numbers?

 Aug 27, 2020
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You have 10 numbers that are being separated into 3 distinct groups. You have to move the 3 groups, in their ordered way, in 3! =6 ways. It is basically the same as moving 3 digits around, such as 123. You would have 3! =6 ways of arranging them: 123, 132, 213, 231, 312, 321. Place the 5 numbers 1, 2, 3, 4, 5 where 1 is if the 6 arrangements of 123. Then place 6, 7, 8 where 2 is. And finally, place 9, 10 where 3 is. Basically, you will have something like this: [12345678910], [12345910678], [67891012345], [67812345910], [91012345678], [91067812345]. 

 Aug 27, 2020

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