Question

There are 3 identical books on English, 4 identical books on Hindi, 2 identical books on Mathematics. In how many distinct ways can they be arranged on a shelf?

Answer 1

Hint: We know that n objects can be arranged in n! ways. If there are identical objects, then we need to divide n! by r!, where r is the number of identical objects. Thus, by using this concept we will get the required number of arrangements.

Complete step by step Answer:

We are given 3 identical books on English, 4 identical books on Hindi, 2 identical books on Mathematics. We need to arrange them on a shelf.
The total number of books = 3+4+2 =9
We also have identical books, so we must consider them while finding the number of arrangements.
Number of ways of arranging n books is given by \[n!\]. Out of n books, if ${r_1},{r_2},{r_3}$ books are identical, the number of ways of arranging is given by,$\dfrac{{n!}}{{{r_1}!\times {r_2}!\times {r_3}!}}$
So, the number of ways the given books can be arranged is ${\text{ = }}\dfrac{{{\text{9!}}}}{{{\text{3!$\times$ 4!$\times$ 2!}}}}$
By doing simplification, we get,
\[
  \dfrac{{{\text{9!}}}}{{{\text{3!$\times$ 4!$\times$ 2!}}}}{\text{ = }}\dfrac{{{\text{9$\times$ 8$\times$ 7$\times$ 6$\times$ 5$\times$ 4!}}}}{{{\text{3$\times$ 2$\times$ 1$\times$ 4!$\times$ 2$\times$ 1}}}} \\
  {\text{ = 1260}} \\
 \]
Thus, the number of ways the books can be arranged is 1260.

Note: We are using the concept of permutations. The equation of permutations for arranging r objects from n objects is ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$. In this case n and r are equal, so the equation reduces to ${}^n{P_n} = \dfrac{{n!}}{{\left( {n - n} \right)!}} = \dfrac{{n!}}{{0!}} = n!$. We are taking permutations and not combinations because while arranging the order of books is important. While doing arrangement problems, we must consider repeating or identical objects. This problem is somewhat similar to the problem of arranging the letters of a given word. For problems involving the arrangement of letters we take n as the total number of letters and also consider the repeating letters. We must know to simplify factorials to find the permutation. $n! = n\times \left( {n - 1} \right)!$ is the basic equation used for simplifying factorials for division.