Given,

Word $\text{'}\mathrm{DISTRIBUTION}\text{'}$

Now, we have $\mathrm{III}, \mathrm{TT}, \mathrm{D}, \mathrm{S}, \mathrm{R}, \mathrm{B}, \mathrm{U}, \mathrm{O}, \mathrm{N}$

Number of words with $3$ alike letter and $1$ different will be,

${=}^8{C}_1\times \frac{4!}{3!}=32$

Number of words with $2$ alike and other $2$ alike will be,

$=\frac{4!}{2!2!}=6$

Number of words with $2$ alike and $2$ different will be,

${=}^2{C}_1{\times }^8{C}_2\times \frac{4!}{2!}=672$

Number of words with all different letters will be,

${=}^9{C}_4\times 4!=3024$

Therefore, the total number of ways will be,

$=3024+672+6+32$

$=3734$