In how many ways five flags of different colors can be arranged if any no. of flags can be taken at a time(it can be 1,2,3,4 or 5)?

The number of ways of arranging:

 

1 flag at a time = ⁵P₁ = 5

2 flag at a time = ⁵P₂ = 5 x 4 = 20

3 flag at a time = ⁵P₃ = 5 x 4 x 3 = 60

4 flag at a time = ⁵P₄ = 5 x 4 x 3 x 2 = 120

5 flag at a time = ⁵P₅ = 5 x 4 x 3 x 2 x 1 = 120

The required number of arrangements =  

                                                 5+20+60+120+120 = 325

 

It can be solved using permutations only.