How many odd numbers between 300 and 400 are there in which none of the digits is repeated ?

Concept:

A permutation is an arrangement of objects in a definite order.

\(\displaystyle ^n P_r=\frac{n !}{(n-r) !}\),

where, 

           ⁿP_r = Permutation

        n = total number of objects

Calculation:

Numbers between 300 and 400 consist of three digits with the digit at hundred’s place equal to 3.

Since repetition is not allowed and there must be only odd numbers, one’s places can be filled using the digits 1,5,7 and 9 in ⁴P₁ ways.  

Now ten’s place can be filled by using any digits except the digit at one's and hundred's place in ⁸P₁ ways. 

Hence, the required number of numbers = 4P1 × 8P1 = 4 × 8 = 32.

∴ The total odd numbers between 300 and 400 in which none of the digits is repeated are 32.