Question

A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of first; what is the total number of signals that can be made?

Answer 1

Hint: Using the concept of permutation and combination as we know that the total there are 5 arms and each arm is capable of 4 distinct positions. And hence we know that there are 5 places and each place can be filled with 4 distinct positions, as not repetition matters. And so, each place can be filled by 4 ways, so on calculating we get the total number of ways.

Complete step by step solution: As we know that a telegraph has 5 arms and each arm is capable of 4 distinct positions.

So, the total five positions \[\left( \_ \right)\left( \_ \right)\left( \_ \right)\left( \_ \right)\left( \_ \right)\]

And one place can be filled with four positions as

\[\left( 4 \right)\left( 4 \right)\left( 4 \right)\left( 4 \right)\left( 4 \right)\]

Hence, the total number of signals that can be made is

\[ \Rightarrow 4 \times 4 \times 4 \times 4 \times 4\]

On calculating above value,

\[ = {4^5} = 1024\]

So, the total number of signals that can be made is 1024.

Note: Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.