Question

How many odd numbers less than 1000 can be formed by using the digits 0, 2, 5 and 7 when the repetition of digits is allowed?

Answer 1

Hint: We will use the concept of multiplication which will be done between the numbers of choices that we will get in each place. This will help in solving the question. For the choices we will consider the digits 0, 2, 5 and 7 only.

Complete step-by-step answer:
According to the question we need to focus on the numbers which are odd and less than 1000. And while forming the digits the numbers we need to consider here are 0, 2, 5 and 7. Now the numbers less than 100 will be a single digit number, a two digit number and can be a three digit number also.
Now we will form places like so which have three places. $-\,,-,\,-$.
Now first we will focus on the last place. This is because we are supposed to find the odd numbers. Therefore, the last digit matters here. As out of the given digits 0, 2, 5 and 7 we have two digits which are odd. These are 5 and 7. Therefore, the last place is going to be filled by any one of these two numbers so that we can get an odd number. This results in two choices that the last place is going to have. Thus, we get $-,-,2$.
Now, we will focus on the first two places one by one. As the repetition is allowed here thus the choices that we are left with now are four only. This results in $4,\_\,,2$. So, we can fill the first place by four given numbers. This means that the first place has 4 choices.
Similarly as there is repetition therefore, the second place also has 4 choices. This gives us $4,4,\,2$.
Now, we will multiply these choices by each other to get the total number of possibilities here. Therefore, we get $4\times 4\times 2=32$.
Hence, the odd numbers less than 1000 formed by using the digits 0, 2, 5 and 7 when the repetition of digits is allowed is given by 32 ways.

Note: We will not apply here the formula of permutation. This is because the permutations are called arrangements and here no arrangements are formed. This is because here repetition is allowed. We will also not use the combination formula. This is because this question is not about questions as we are forming numbers and not combining them. Combination is mostly used in case of letter formations. Before starting with such a question one needs to focus on the important points that are important for solving questions. Here the important points are given here. First is that the number that is going to be formed here is not only a three digit number but can be a two and single digit number. Another important point is that we need to focus only on the digits which are given in the question, otherwise we will form wrong numbers also.