Question

How many 3 digit numbers are there, with distinct digits, with each of the digits odd?

Answer 1

Hint: We will first create three spaces which will denote the 3 digits. Then we will list the odd numbers and put them into those spaces taking care that we do not repeat them as they need to be distinct digits.

Complete step-by-step answer:
Let us begin with the solution.
Now there are 5 odd numbers from 0 to 9. Those are 1,3,5,7 and 9. Now we need to fill these into three places and make sure that the numbers do not repeat.
Let us take the example of the number 135.

1st digit	2nd digit	3 digit
1	3	5

Now 135 is the first number which satisfies all the conditions.
Similarly, we can find many numbers but this would take a lot of time.
Let us generalize this.
Now we have 5 options to be put in the 1st digit place. So
1st place = 5 options.
Now since the numbers do not have to repeat the number of options we need to put in the second place will reduce to (5-1) that is 4. Thus, 2nd place = 4 options.
Now in the third place too the numbers should not repeat. Thus our options further reduce to (5-3) that is 3 options. Thus,
3rd place = 3 options.
Now, to find all the arrangements, we multiply the number of options in all places.
Thus the total number of numbers is equal to \[5\times 4\times 3=60\].
Thus there are 60, 3 digit numbers, with distinct digits, with each of the digits odd.

Note: We should be very careful about multiplying those numbers. Since it is told that the numbers are distinct the options should be one less than the previous one. If we are multiplying 5 for every digit then we are committing a mistake and we will get a wrong answer. Also, remember we should multiply those numbers and not add them as the number is formed by all the three digits and not with any of the three numbers.