Question 1187402
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            So,  I  do not touch part  (1)  and am focusing on part  (2),  merely.



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Imagine 8 different English books on the shelf.


Assume that they are not tightly placed each to other, having gaps between them.


We should place Math books into these gaps:  one (and no more than one) Math book into each of 7 gaps.


Add to it one potential place on the left of 8 English books and one potential place on the right of 8 English books.


So, you have 7+1+1 = 9 potential places for 4 Math books.


It explains this factor  {{{C[9]^4}}}, which you see in the formula.


Also, 8 English books can be ordered in 8! ways. 

It explains the factor 8! in the formula.



Now, the numerator  8!*C[9]^4  is the number of all favorable dispositions.

The denominator 12! = (8+4)!  is the number of ALL potential placements of the books on the shelf.


The ratio of  {{{favorable}}}  to  {{{total}}},  {{{favorable/total}}},  is the probability, which you are seeking for.
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Do you understand my explanation/explanations in full ?