Question

Felicia has three best friends named Bob, Cassandra, and Hubert. How many ways are there to line up the eight kids so that Felicia is next to at least one of her three best friends?

Answer 1

Total ways are there to line up the eight kids so that Felicia is next to at least one of her three best friends are

If Felicia at start or end position then there will be only two place where her one friend can stand

so select one place as ²C₁

and select one friend from three as ³C₁

so one place of Felicia from 2 place and one place for her friend fixed and remain 6 place can be fill as 6! type

so ways will be=²C₁ * ³C₁ * 6!=2*3*720=4320

If Felicia at any position(except start or end) then there will be two place (left or right)where her one friend can stand

so select one place for Felicia from remain six place as ⁶C₁

and select one friend from three as ³C₁

two place (left or right)where her one friend can stand as ²C₁

so one place of Felicia from 6 place and one place for her friend fixed and remain 6 place can be fill as 6! type

so ways will be=⁶C₁ * ²C₁ * ³C₁ * 6!=6*2*3*720=25920

So total ways are =4320+25920=30240