How many odd numbers less than 1000 can be formed from the digits 0, 1, 8, 9 if the repetitions are allowed.

Hint: Here we have to find the number of odd numbers that can be formed using the digits 0, 1, 8, 9. Numbers less than 1000 will be 3 digit numbers, 2 digit numbers and 1 digit numbers. For 3 digit odd numbers, we have to find the number of permutations of four digits taking three at a time but we can’t take 0 at hundredth place and unit place and also we can’t take 8 at the unit’s place. Again for 2 digit numbers, we have to find the number of permutation of four digits taking two at a time but we can’t take 0 at tenth place and unit’s place and also we can’t take 8 at the unit’s place and similarly for single digit numbers, we have to find the number of permutation of four digits taking one at a time except 0 and 8.

Complete step by step solution:
We have to find the odd numbers less than 1000. The odd numbers less than 1000 will be of 3 digit, 2 digit and single digit number.
First, we will find the number of 3 digit odd numbers.
Number of digits placed at unit’s place $ = 2$ as we can’t include 0 and 8 here
Number of digits placed at tenth’s place $ = 4$ as repetitions of digits are allowed.
Number of digits placed at hundred’s place $ = 3$ as we can’t include 0 here
Thus, the total number of 3 digit odd numbers$ = 3 \times 4 \times 2 = 24$
Now, we will find the number of 2 digit odd numbers.
Number of digits placed at unit’s place $ = 2$ as we can’t include 0 and 8 here
Number of digits placed at tenth’s place $ = 3$ as we can’t include 0 here.
Thus, the total number of 2 digit odd numbers$ = 3 \times 2 = 6$
Now, we will find the number of single digit odd numbers.
Number of digits that we can include in single digit odd numbers is 2
Hence, total number of odd numbers less than 1000$ = 24 + 6 + 2 = 32$.

Note: We need to know the meaning of the factorial because sometimes we use a permutation formula which is the ratio of the factorials.
(i)Factorial of any positive integer is defined as the multiplication of all the positive integers less than or equal to the given positive integers.
(ii)Factorial of zero is one.
(iii)Factorials are commonly used in permutations and combinations problems.
(iv)Factorials of negative integers are not defined