Question

How many ways are there to arrange the letters ‘a’, ‘b’, ‘c’, ‘d’, and ‘e’ such that ‘a’ is not immediately followed by ‘e’ (no repeats since it is an arrangement)? Justify your answer using the product rule, the sum rule, and/or the subtraction rule .

Answer 1

We need to arrange the letters a , b , c , d , e such that a is not immediately followed by e .

If we fix the letter a in first position then is is not followed by any letters so in rest of the four place remianing four letter can be arrange arbitrarily i.e., there will be no restriction . So total number of possibilities in ths case is ,

= 24

If we fix athe letter a in second place then first letter cannot be e so first leter wil have 3 possibilities and the rest of the three place have no restriction so there will be 3! possibilities for rest of the three letters . So total number of possibilities in this case is ,

= 18

If we fix the letter a in third position then ae cannot be in second position so second position will have 3 possibilities and ret of the three places will have 3! possibilities . Hence toltal number of possibilities here is ,

=

= 18

Similiarly if we fix a in fourth and fifth position then it will have 18 possibilities .

Hence number of way they can be arrange is ,

Hence ,

Answer : 96