To find the number of $7$ digit odd number formed by using the seven digits $1,2,2,2,3,3,5$

We will take following cases,

Case $1-$ fixing $1$ as unit place,

So total number of ways will be $\frac{6!}{3!2!} \left\{\text{as }2 \text{is repeating three times and 3 is repeating two times}\right\}$

Case $2-$ fixing $3$ unit place,

So total number of ways will be $\frac{6!}{3!}$

Case $3-$ fixing $5$ as unit place,

So total number of ways will be $\frac{6!}{3!2!}$

Now adding all the cases we will get, total number of ways as $\frac{6!}{3!2!}+\frac{6!}{3!}+\frac{6!}{3!2!}=60+120+60=240$