Q.If the letters of the word 'REGULATIONS' be arranged at random, what is the probability that there will be exactly 4 letters between $R$ and $E$?

REGULATIONS has 11 distinct letters (R,E,G,U,L,A,T,I,O,N,S - no repeats), $n(S)=11!$. 'Exactly 4 letters between R and E' means their positions differ by exactly $4+1=5$. Among positions 1 to 11, pairs of positions 5 apart: $(1,6),(2,7),(3,8),(4,9),(5,10),(6,11)$ - 6 unordered position-pairs, each usable in 2 orders (R before E, or E before R): 12 ordered position-pairs. For each, the remaining 9 letters fi …

Count position-pairs at the required distance apart (times 2 for R-then-E or E-then-R), multiply by the arrangements of the r …

Using a distance of 4 instead of 5 for 'exactly 4 letters between' - 4 letters between two position …