MEDGAR ENERSCOLLEGEDEPARTMENT oF MATHEMATIcs MITH 27 FALL 20

MEDGAR ENERSCOLLEGEDEPARTMENT oF MATHEMATIcs MITH 27, FALL 2016 FINALEKAMPROF TERRENCE BLACMw Problem 8. (20 points) How many different linear arrangements are there of the letters A, B, C,D, E, F for which (a) A and Bare next to each other? (b) A is before B? (c) Ais before B and B is before C? (d) Ais before B and Cis before D? (e) and B are next to each other and Cand Dare also next to each other? (fEis not last in line?

Solution

A and B are next two each other

take AB as one entity

now how many arrangements? AB, C, D , E , F

5! = 120 , Also A and B can interchange

so 5! *2 = 120*2 = 240

2) A is before B

out of all the arrangements which are 6! =720

in exactly 1/2 A will be before B . So the answer = 360

in other way 720/2! is the answer since order f AB is fixed

3) A before B and B before C

order of ABC is fixed

hence 6!/3!= 120 is the answer

4) A is before B we have 360

in half of them C is before D and in ther half D is before C

So the answer is 360/2 = 180