Q.In how many ways can 5 different books be arranged on a shelf so that two particular books are never together?

EXPRESSION has 10 letters with $E,E$ and $S,S$ repeated (6 other distinct letters); split the 4-letter selections into 3 cases by how many repeated pairs are used, then count combinations and their arrangements separately.

Letters of EXPRESSION: $E,X,P,R,E,S,S,I,O,N$ — 10 letters total, with $E$ appearing twice, $S$ appearing twice, and $X,P,R,I,O,N$ each appearing once (6 distinct singles).

Combinations (selections of 4 letters), by case:

Case 1 — both pairs used ($E,E,S,S$): only $1$ way to choose this multiset.

Case 2 — exactly one pair used (either $EE$ or $SS$) plus 2 distinct singles from the 6: choose the pair in $2$ ways, choose 2 singles in $^6C_2=15$ ways: $2\times15=30$ combinations.

Case 3 — no pair used, all 4 letters distinct, chosen from the 8 distinct letter-types ($E,X,P,R,S,I,O,N$): $^8C_4=70$ combinations.

Total combinations $=1+30+70=101$.

Permutations (arrangements) for each case:

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