Question 488859
Without ordering the books of the same color, 
here is the possible ways to order books of different colors:
(1) [ 4 red ][ 3 green ][ 3 gray ]
(2) [ 4 red ][ 3 gray ][ 3 green ]
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(3) [ 3 green ][ 4 red ][ 3 gray ]
(4) [ 3 gray ][ 4 red ][ 3 green ]
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(5) [ 3 gray ][ 3 green ][ 4 red ]
(6) [ 3 green ][ 3 gray ][ 4 red ]
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So, there are 6 different ways to order
books of different colors (1) through (6)
The possible ways that books of the same color
can be ordered for each of the (6) is:
{{{ 4! * 3! * 3! = 24*6*6 }}}
{{{ 24*6*6 = 864 }}}
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You can have (1) OR (2) OR (3) OR (4) OR (5) OR (6)
So, the choices for different color orderings get added
{{{ 864 + 864 + 864 + 864 + 864 + 864 = 5184 }}}
There are 5184 possible arrangements of the books
You should get other opinions, too.