A librarian has a set of ten different books, including four books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible?
(A) (10!)/(4!)
(B) (4!)(6!)
(C) (4!)(7!)
(D) (4!)(10!)
(E) (4!)(6!)(10!)
Source: Magoosh
$$?\,\,\,:\,\,\,\# \,\,\,{\rm{possibilities}}\,{\rm{,}}\,\,{\rm{Abe}}\,\,{\rm{books}}\;\,{\rm{together}}$$
$$10\,\,{\rm{different}}\,\,{\rm{books,}}\,\,{\rm{4}}\,\,{\rm{about}}\,{\rm{Abe}}\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{
\,1\,\,{\rm{multiple - block}}\,\,\,\left( {4\,\,{\rm{Abe}}\,\,{\rm{books}}} \right) \hfill \cr
\,6\,\,{\rm{single}}\,{\rm{blocks}} \hfill \cr} \right.$$
$$\left. \matrix{
{P_7} = 7!\,\,\,{\rm{permutation}}\,\,{\rm{of}}\,\,{\rm{all}}\,\,{\rm{blocks}} \hfill \cr
{{\rm{P}}_{\rm{4}}} = 4!\,\,\,{\rm{permutation}}\,\,{\rm{of}}\,\,{\rm{Abe}}\,\,{\rm{books}}\,\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\, = \,\,{P_7} \cdot {P_4}\,\, = \,\,7!4!\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{C}} \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
