Question

A library has ‘a’ copies of one book, ‘b’ copies each of the two books, ‘c’ copies each of three books, and a single copy of d books. The total number of ways in which these books can be arranged in a shelf is equal to
A. $\dfrac{\left( a+2b+3c+d \right)!}{a!{{\left( b! \right)}^{2}}{{\left( c! \right)}^{3}}}$
B. $\dfrac{\left( a+2b+3c+d \right)!}{a!\left( 2b! \right){{\left( c! \right)}^{3}}}$
C. $\dfrac{\left( a+b+3c+d \right)!}{{{\left( c! \right)}^{3}}}$
D. $\dfrac{\left( a+2b+3c+d \right)!}{a!\left( 2b! \right)\left( 3c! \right)}$

Answer 1

In this problem, we have to find the total number of ways in which these books can be arranged in a shelf equal to which of the given options. We should know that, the number of ways of arranging n unlike objects in a line in n! for example, the different ways the a, b, c, d can be arranged. The number of ways of arranging n objects of which a of one type are alike, b of second type are alike and c of third type are alike is $\dfrac{n!}{a!b!c!}$, where $n=a+b+c+d$.

Complete step by step solution:
We know that the given data is,
A library has ‘a’ copies of one book, ‘b’ copies each of the two books, ‘c’ copies each of three books, and a single copy of d books.
We can write this above statement in an equation as,
The total number of books, n = $a+2b+3c+d$ .
The number of ways of arranging n objects of which a of one type are alike, b of second type are alike and c of third type are alike is $\dfrac{n!}{a!b!c!}$.
Now we can write the equation as,
$\Rightarrow \dfrac{\left( a+2b+c+d \right)!}{a!{{\left( b! \right)}^{2}}{{\left( c! \right)}^{3}}}$
Therefore, the option A. $\dfrac{\left( a+2b+3c+d \right)!}{a!{{\left( b! \right)}^{2}}{{\left( c! \right)}^{3}}}$ is the correct answer.

Note: We should always remember that the number of ways of arranging n objects of which a of one type are alike, b of second type are alike and c of third type are alike is $\dfrac{n!}{a!b!c!}$ where,$n=a+b+c+d$. We should also concentrate on the part of forming an equation for the total number of books from the given data.