A telegraph has x arms and each arm is capable of \[(x - 1)\] distinct position, including position of rest. The total number of signals that can be made is:

Hint: Here we use the given information about the number of distinct positions of each arm. Using the property of exponents to write the number of signals formed and exclude the signal that indicates the position of rest.
* If a number ‘a’ is multiplied to itself ‘n’ times then, \[\underbrace {a \times a \times ...... \times a}_n = {a^n}\].

Complete step-by-step answer:
A telegraph is a communication device that was used to transmit messages along wire, by creating signals.
We are given the number of arms in a telegraph is ‘x’.
Number of distinct positions each arm can have is \[(x - 1)\].
\[ \Rightarrow \]First arm can take any of the \[(x - 1)\] distinct positions.
Similarly, the second arm can also take any of the \[(x - 1)\] distinct positions.
We keep moving in the same order; the \[{x^{th}}\] arm can take any of the distinct \[(x - 1)\] positions.
Total number of signals is given by multiplying the number of signals obtained from each arm.
\[ \Rightarrow \]Total number of signals \[ = \underbrace {(x - 1) \times (x - 1) \times ...... \times (x - 1)}_x\]
Use the law of exponents which states that \[\underbrace {a \times a \times ...... \times a}_n = {a^n}\]
\[ \Rightarrow \]Total number of signals \[ = {(x - 1)^x}\]...........… (1)
Now we have to remove the position of rest from the total number of signals as that position does not give any signal. Subtract 1 from equation (1)
\[ \Rightarrow \]Total number of signals \[ = {(x - 1)^x} - 1\]

\[\therefore \]Total numbers of signals that can be made are \[{(x - 1)^x} - 1\].

Note: Students might make the mistake of writing the number of signals formed by adding the number of signals formed from each arm. Remember that when we are referring to the number of signals formed we are referring to the number of different positions of each arm, so on fixing any one of the positions we have many different choices for other positions, So we take product of the number of signals made by each arm.