Question

How many odd number less than 10000 can be formed using the digits 0,2,3,5 . Repetation is allowed?

Answer 1

We have to determine the numbers less than 10,000 can be formed from digits 0,2,3,5

Since the numbers should be less than 10,000. So, it can not be five digit number.

It can be four digit number, three digit number, two digit number and one digit number.

The number of ways in which four digit numbers are formed:

The thousands place can be formed in 3 ways(2,3,5) = 3

The hundreds place can be formed in 4 ways (0,2,3,5) = 4

The tens place can be formed in 4 ways (0,2,3,5) = 4

The units place can be formed in 2 ways (3,5) as odd numbers are formed = 2

So, the total number of ways to form four digit numbers = $3 \times 4 \times 4 \times 2 = 96$.

The number of ways in which three digit numbers are formed:

The hundreds place can be formed in 3 ways (2,3,5) = 3

The tens place can be formed in 4 ways (0,2,3,5) = 4

The units place can be formed in 2 ways (3,5) as odd numbers are formed = 2

So, the total number of ways to form three digit numbers = $3 \times 4 \times 2 = 24$.

The number of ways in which two digit numbers are formed:

The tens place can be formed in 3 ways (2,3,5) = 3

The units place can be formed in 2 ways (3,5) as odd numbers are formed = 2

So, the total number of ways to form two digit numbers = $3 \times 2 = 6$.

The number of ways in which two single number are formed:

The units place can be formed in 2 ways (3,5) as odd numbers are formed = 2

So, the total number of ways to form single digit number =2.

So, the total numbers formed = 96+24+6+2 = 128.