The requirement in the task that in any position of the pipe, at least one outlet must be in the hole with the same number, is equivalent to the requirement that exactly one outlet must be in the hole with the same number. Indeed, if in some position of the pipe, 2 or more outlets were in the corresponding hole, then there would be a position where none of them would be in the correct hole.

We can assume that the outlets are numbered in the direction of the clock hands as $1, 2, 3, 4, 5$. Let's count how many ways the holes can be numbered. Place the pipe in the fitting, and where the 1st outlet is, let that be the 1st hole. By rotating the pipe in the direction of the clock hands by $72^{\circ}$, the 5th outlet will be in the 1st hole. In this position, either the 2nd outlet (Case I), or the 3rd (Case II), or the 4th outlet (Case III) can be in its place.

Case I: If in this position we number the hole where the 2nd outlet is with the number 2, then the next hole, where the 3rd outlet is now, can only get the number 5. Indeed, in this hole, when the 1st number was in place, the 4th outlet was there, and when the 2nd number was in place, the 3rd number outlet was there, so it cannot be 1, 2, 3, or 4. By further rotating the pipe, again by $72^{\circ}$, the 3rd and 5th outlets will go to unnumbered holes. Therefore, the numbering is uniquely determined: in this position, the 3rd outlet can only be in its place. Consequently, in the direction of the clock hands, in Case I, we only get the numbering 1, 4, 2, 5, 3.

Similarly, by examining Case II, we only get the numbering 1, 5, 4, 3, 2, and in Case III, we only get the numbering 1, 3, 5, 2, 4. Therefore, the fitting can be numbered in 3 different ways.
A radio tube has 5 connections and its socket has 5 holes. These are located at the vertices of a regular pentagon, and the tube can be placed in the socket in 5 different ways. The connections are numbered from 1 to 5. In how many ways can the holes be numbered from 1 to 5 so that in every position of the tube at least one connection is in the hole with the same number?