Question

There are 2 identical white balls, 3 identical red balls and 4 green balls of different shades. The number of ways in which they can be arrange in a row so that at least one ball is separated from the balls of the same colour, is

A. $$6\left(7!-4!\right)$$
B. $$7\left(6!-4!\right)$$
C. $$8!-5!$$
D. None

Answer 1

Hint: The number of ways balls are arranged in a row so that at least one ball is separated from the balls of the same colour is given by the difference of the total number of ways that the balls can be arranged and the number of ways all identical balls can be arranged together. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:

Given number of identical white balls = 2
Number of identical red balls = 3
Number of green balls = 4
So, the total number of balls = 2 + 3 + 4 = 9
Out of the total 9 balls, 2 are identical of one kind, 3 are alike of another kind and 4 are distinct ones.
Number of ways balls are arranged in a row so that at least one ball is separated from the balls of same colours = Total number of ways – all identical balls together.
So, total number of ways balls can be arranged is given by
$$\begin{gathered} \Rightarrow n\left(S\right)={\displaystyle \frac{9!}{2!3!}} \\ \Rightarrow n\left(S\right)={\displaystyle \frac{9!}{\left(2\right)\left(6\right)}} \\ \Rightarrow n\left(S\right)={\displaystyle \frac{9\times 8\times 7!}{12}} \\ \therefore n\left(S\right)=6\left(7!\right) \end{gathered}$$
And the number of ways balls are arranged so that identical balls are together is given by
$$\begin{gathered} \Rightarrow n\left(T\right)=3!\times 4! \\ \Rightarrow n\left(T\right)=3\times 2\times 1\left(4!\right) \\ \therefore n\left(T\right)=6\left(4!\right) \end{gathered}$$
Hence, the number of ways balls are arranged so that at least one ball is separated is given by
$$\Rightarrow n\left(S\right)-n\left(T\right)=6\left(7!\right)-3\left(4!\right)=6\left(7!-4!\right)$$
Thus, the correct option is A. $$6\left(7!-4!\right)$$

Note: An arrangement (or ordering) of a set of objects is called a permutation (we can also arrange just part of the set of objects). In permutation, the order that we arrange the objects in is important.