1.
PERMUTATIONS there are always 4 letters between P & S
P[4 letters]S and this lump can be placed in the 7 gaps created by arrangement of the leftover 6 letters.
=(10C4*!4) *(!6 * 7)/!2
[ here !2---for the two T's & leaving P&S, 4 letters can be selected and distributed in 10C4*!4]
since P&S can be arranged among themselves in two ways
so =(10C4*!4) *(!6 * 7)=(!10*!4*!6*7)/(!6*!4)
=!10*7=25,401,600
2.
MISSISSIPPI where the 4 I's never come together
this is the simplest case: total arrangements of the given word
MINUS the arrangements where the 4 I's are together
=!11/(!4*!4*!2) - !8/(!4*!2)
=!8/(!4*!2)[(11*10*9)/24 - 1]
=(8*7*6*5*161)/8=33810