Question

How many ways can 6 different books be arranged on a shelf?

Answer 1

To solve the above question we will use the term factorial (!). Actually factorial is a function that multiplies a number by every number below it. For example, we have $5!$ , then we can write it as $5!=5\times 4\times 3\times 2\times 1$. This factorial function is used among other things, to find the number of ways “n” objects can be arranged. This factorial term is used in many mathematical formulas such as permutation and combination.

Complete step by step solution:
Now in the above question we have to find the number of ways by which six books can be arranged on a self.
In mathematics, there are n! ways of arranging n distinct objects into an ordered sequence. The factorial n! gives the number of ways in which n objects can be permuted. For example, $2$ factorial is $2!=2\times 1$ , it means there are two different ways to arrange the numbers $1$ through $2$ which are\left\\{ 1,2 \right\\},\left\\{ 2,1 \right\\}.
Now to arrange six books on a shelf, that means we have six spots to put them in.
Our first book has six spots it can be in, and then our second book has $5$ , since one is taken by the first book. Then the next will have $4$, and then $3$ , then$2$ , then $1$. Thus we can write it as:
$\Rightarrow 6!=6\times 5\times 4\times 3\times 2\times 1=720$
Hence we have $720$ ways to arrange the books on shelf.

Note: The above question is simple because here is an arranged number on a shelf. But sometimes we have to arrange the letters in a circular form, so for the circular form the number of arrangements of elements is$\left( n-1 \right)!$.