Worked solution
Arrange PETALS with restrictions on P and E
Find the number of ways to arrange the six letters of the word PETALS when (i) there is no restriction, (ii) P is the first letter and E is the last letter, (iii) P and E are together, (iv) P and E are separated.
CJC · Lecture · Self-Practice 5
(i) Arrange six distinct letters
There are arrangements.
(ii) Fix both end letters
With P first and E last, arrange the other four distinct letters in the middle: .
(iii) Make a block containing P and E
Arrange the block and T,A,L,S as five outer objects, then count PE and EP inside: .
(iv) Subtract adjacent arrangements
Separated means not adjacent. Every arrangement is either adjacent or separated, giving . The four printed answers are 720,24,240,480; these derivations supplement the answer-only Self-Practice source.
Common mistakes
P and E may occur in either internal order when together. For separated letters use the complement of adjacency, not a fixed distance.