The number of seven digits odd numbers, that can be formed using...

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 $\dfrac{6 !}{3 ! 2 !}\left\{\text{as}2\text{is repeating three times and 3 is repeating two times}\right\}$

Case $2 -$ fixing $3$ unit place,

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

Case $3 -$ fixing $5$ as unit place,

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

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