The number of four letter words that can be formed using the letters of the word $BARRACK$ is 
 

$\mathrm{In} \mathrm{the} \mathrm{word} \mathrm{BARRACK}$

$\mathrm{There} \mathrm{are} \mathrm{RR}, \mathrm{AA}, \mathrm{B}, \mathrm{C}, \mathrm{K}.$

$\mathrm{Number} \mathrm{if} 4 \mathrm{letter} \mathrm{words} \mathrm{having}$

$\left(\mathrm{i}\right) \mathrm{all} 4 \mathrm{district} \mathrm{letters} \left\{\mathrm{R},\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{K}\right\}=5{\mathrm{p}}_4=120$

$\left(\mathrm{ii}\right) 2 \mathrm{pairs} \mathrm{of} \mathrm{identified} \mathrm{letters} \boxed{\mathrm{R}}\boxed{\mathrm{R}}\boxed{\mathrm{A}}\boxed{\mathrm{A}} =\frac{4!}{2!2!}=6$

$$\begin{gathered} \left(\mathrm{iii}\right) 2 \mathrm{idential} 2 \mathrm{different} =2\times 4{\mathrm{C}}_2\times \frac{4!}{2!} \\ \boxed{\mathrm{R}}\boxed{\mathrm{R}}\boxed{ }\boxed{ } \\ \mathrm{A},\mathrm{B},\mathrm{C},\mathrm{K} =6\times 24=144 \\ \boxed{\mathrm{A}}\boxed{\mathrm{A}}\boxed{ }\boxed{ } \\ \mathrm{R},\mathrm{B},\mathrm{C},\mathrm{K} \end{gathered}$$

$\mathrm{Total} \mathrm{number} \mathrm{of} 4 \mathrm{letter} \mathrm{words}=120+6+144=270$