Question

Between two ends of a bookshelf in your study are displayed five of your favourite puzzle books. You wish to place them in all possible combinations. It takes a minute to list each combination. How much time would it take to list all combinations?
[a] One hour
[b] Two hours
[c] Three hours
[d] Four hours

Answer 1

Hint: Use fundamental principle of counting. According to fundamental principle of counting if a task A can be done in m ways and another task B can be done in n ways then the number of ways of doing both the tasks is mn and the number of ways of doing either of the tasks is m+n. Hence determine the total possible arrangements of the books and hence determine the total time taken to arrange the books in all possible ways.

Complete step by step answer:
Let us denote the five places for the five books as P1, P2, P3, P4 and P5.
Now P1 can be filled in 5 different ways (five possible books)
P2 can be filled in 4 different ways (four choices only as we have selected one book for P1)
P3 can be filled in 3 different ways (We are left with only three books)
P4 can be filled in 2 different ways (Only two choices left)
P5 can be filled in 1 way (The remaining book)
We know by fundamental principle of counting if a task A can be done in m ways and another task B can be done in n ways then the number of ways of doing both the tasks is mn and the number of ways of doing either of the tasks is m+n
Hence the number of possible arrangements of the books is $5\times 4\times 3\times 2\times 1=120$
Since it takes 1 min for shifting from one arrangement to other, the total time taken to form all possible permutations assuming no permutation is repeated is $\dfrac{120}{1}=120\min =2hrs$
Hence the total time taken is 2 hours

So, the correct answer is “Option B”.

Note: [1] Alternative solution:
We know that the number of possible arrangements of n different things taken n at a time is given by $n!$
Hence the number of possible arrangements of the five different books is $5!=120$ which is the same as obtained above.
Proceeding as the above solution, we get option [b] is correct.