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Question: In a watch hop there are \[14\] types of wall clocks. Each type has four pieces. The number of ways ...

In a watch hop there are 1414 types of wall clocks. Each type has four pieces. The number of ways in which they can be arranged in a row is:
A. 56!14!\dfrac{{56!}}{{14!}}
B. 56!(14!)4\dfrac{{56!}}{{{{\left( {14!} \right)}^4}}}
C. 56!(4!)14\dfrac{{56!}}{{{{\left( {4!} \right)}^{14}}}}
D. 56!(13!)3\dfrac{{56!}}{{{{\left( {13!} \right)}^3}}}

Explanation

Solution

In order to solve the above question, we will use permutations. Permutation refers to an arrangement of the members of a set into a sequence. It is an act of changing the linear order. We will be using two formulas to solve this question. The first formula will be used to calculate the number of total wall clocks and the second formula will be for permutation.

Formula used: The two formulas that we will be using to solve the above question are:
The first formula is for calculating the number of wall clocks:
Total wall clocks = types of wall clocks × number of pieces of each clock{\text{Total wall clocks = types of wall clocks }} \times {\text{ number of pieces of each clock}} .
The second formula that we will be using is for calculating permutations: P(n,r)=n!(n−r)!{\text{P}}\left( {n,r} \right) = \dfrac{{n!}}{{\left( {n - r} \right)!}} .

Complete step by step solution:
We have to calculate the number of ways in which wall clocks can be arranged in a row.
First, we have to calculate the number of wall clocks. For this we will use the first formula.

Total wall clocks = types of wall clocks × number of pieces of each clock  = 14×4  = 56  {\text{Total wall clocks = types of wall clocks }} \times {\text{ number of pieces of each clock}} \\\ {\text{ = }}14 \times 4 \\\ {\text{ = 56}} \\\

So, we know that the total number of wall clocks are 5656 .
Now, we have to use the formula for permutations to determine the final answer.
We know, P(n,r)=n!(n−r)!{\text{P}}\left( {n,r} \right) = \dfrac{{n!}}{{\left( {n - r} \right)!}} .

Number of permutations are = 56!4!4!4!.....4!  = 56!(4!)14  {\text{Number of permutations are = }}\dfrac{{56!}}{{4!4!4!.....4!}} \\\ {\text{ = }}\dfrac{{56!}}{{{{\left( {4!} \right)}^{14}}}} \\\

So, the correct answer is Option C.

Note: Remember that the use of formulas is very important while solving sums similar to the above question. Here we have used the concept of permutation which means an act of changing the linear order. We have also used another formula, to calculate the total number of wall clocks.