Question

There are counters available in $3$ different colours (atleast four of each colour). Counters are all alike except for the colour. If $m$ denotes the number of arrangements of four counters if no arrangement consists of counters of same colour and $n$ denotes the corresponding figure when every arrangement consists of counters of each colour, then ?

• A. $m=2n$
• B. $6m=13n$
• C. $3m=5n$
• D. $5m=3n$

Answer 1

The correct option is B $6m=13n$

let $4$ counters be $C_1{C}_2{C}_3{C}_4$

Counter colours available $=3$

$\frac{C_1}{3}\frac{C_2}{3}\frac{C_3}{3}\frac{C_4}{3}$

(i) Each counter can be placed with $3$ different colours ,

$m=3^4-3=78$

(ii)$\frac{C_1}{2}\frac{C_2}{2}\frac{C_3}{2}\frac{C_4}{2}$

Select any colours $={}^3{C}_2(2^4-2)$

$n=3^4-{}^3{C}_2(2^4-2)-3=36$

$\frac{m}{n}=\frac{78}{36}\Rightarrow 6m=13n$