Question

There are $5$ white,$4$ yellow,$3$ green,$2$ blue and $1$ red balls. All the balls are identical except for the color. These are to be arranged in a line in $5$ places. Find the number of distinct arrangements.

• A. $2322$
• B. $5230$
• C. $1244$
• D. $2111$

Answer 1

Answer: (D)

$2111$

$5⟶$ White

$4⟶$ Yellow

$2⟶$ Blue

$1⟶$ Red

Cases

(1) All are diffrent

(2) A pair and $3$ diffrent

(3) $2$ Pair of diffrent colors and $1$ diffrent colour

(4) A pair and triad of diffrent colours

(5) A triad of $1$ colour and $2$ singles of diffrent colours

(6) A quad of single colour and a singles of another colour

(7) All five balls of same colour

$\Rightarrow \left(1\right)+\left(2\right)+\left(3\right)+\left(4\right)+\left(5\right)+\left(6\right)+\left(7\right)$

$\Rightarrow 5!{+}^4{C}_1{\times }^4{C}_3\times \frac{5!}{2!}$${+}^4{C}_2{\times }^3{C}_1\times \frac{5!}{2!\times 2!}$$+{\left({}^3{C}_1\right)}^2\times \frac{5!}{2!\times 3!}$${+}^3{C}_1{\times }^4{C}_2\times \frac{5!}{2!}{+}^2{C}_1{\times }^4{C}_1\times \frac{5!}{2!}+1$

$=2111$