Question

A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of rest. Find the total number of signals that can be made.

Answer 1

Telegraph is an instrument which has $n$ number of arms and capable of $m$ distinct positions. The telegraph produces the signals based on the position of the arms. For a distinct position of arms, there is an individual signal produced. So, the $n$ number of arms can arrange in $m$ distinct places in ${{m}^{n}}$ ways. If the rest position is included in ${{m}^{n}}$ ways then the number of signals produced is ${{m}^{n}}-1$.

Complete step-by-step solution
Given that, the telegraph has $5$ arms and each arm is capable of $4$ distinct position then the number of ways that the $5$ arms can be arranged in $4$ distinct positions is ${{4}^{5}}$. In the problem they mentioned the 4 distinct positions including the position of rest, so we need to subtract the rest position from the number of possible positions. Then the possible number of positions is ${{4}^{5}}-1$. Hence the number of signals produced by the telegraph in ${{4}^{5}}-1$ arrangements of arms are possible with the help of $5$ arms and each arm are capable of $4$ distinct position is
$\begin{aligned} & {{4}^{5}}-1=1024-1 \\\ & =1023 \end{aligned}$

Note: Remember that in the question they mentioned that the rest position is also included in the possible positions then we have to subtract it from the number of possible positions. So many times, students don’t know the actual concept of the telegraph so they multiple $5$ with $4$ and write the answer as $20$. So, remember that for each possible position of the arm there is one signal, so we need to find the total number of possible positions of arms to get the number of signals.