A shopkeeper sells three varieties of fruit juice. He has a large number of bottles of same size of each variety. The number of different ways of displaying all the three varieties on the shelf with 5 places in a row and each display must have at least one bottle of each variety is

Step-by-step solution

Let the three varieties of fruit juice be A , B , and C . We need to fill 5 places in a row such that each variety appears at least once. This is equivalent to finding the number of onto functions from a set of 5 elements to a set of 3 elements. The total number of ways to fill 5 places with 3 varieties without any restriction is 3^5 = 243 . Let S be the set of all possible arrangements. Let P₁, P₂, P₃ be the properties that variety A, B, C are missing respectively. We want to find the number of arrangements that s