The  number of permutations of the letters of the word ‘ENGINEERING’  in which  all “E”s are not together and all  N’s are  together  is

NNN = One unit Remaining =8 units. Total 9 units 
$\therefore$  Total permutations – E’s come together

$\text{ Total permutation }=\frac{9!}{3!2!2!}$

$\text{ Permutations when E's comes together }=\frac{7!}{2!2!}$

if all “E”s are not together and N’s come together then number of permutations is $\frac{9!}{3!2!2!}-\frac{7!}{2!2!}$