Answer to Question #306324 in Discrete Mathematics for kasish

Question

5. You have 5 different-colored bottles, each with a distinct cap. In how many ways can these caps be put on the bottles such that none of the caps are on the correct bottles? (Assume that all the caps must be on the bottles.) 

Answer 1

Solution

If we are asked not to put the caps on the correct bottle, then

The first bottle has 4 ways (excluding the correct cap)

The second bottle has 3 ways (excluding the correct cap + already used cap)

The third bottle has 2 ways

The fourth bottle has 1 way, and

The fifth bottle has 1 way.

Therefore,

The total number of ways are:

$4×3×2×1×1 = 24$