Question

How many numbers less than 10000 can be made with the eight digits 1, 2, 3, 0, 4, 5, 6, 7?

Answer 1

Hint: Use the fact that the numbers less than 10000 can have 4 digits, 3 digits, 2 digits and 1 digit. Count all possible numbers of each of the kind keeping in mind that repetition of digits is allowed. Add all the values obtained to get the total count of numbers which can be formed.

Complete step-by-step answer:
We have to count all the numbers less than 10000 which can be made with the digits 1, 2, 3, 0, 4, 5, 6 and 7.
We observe that the numbers less than 10000 can have 4 digits, 3 digits, 2 digits and 1 digit. We will count all the possible numbers of each kind.
Firstly, we will consider the 1 digit numbers.
We can place any one of the 8 digits to form a one digit number.
Thus, the total number of 1 digit numbers which can be formed are 8.
We will now consider the 2 digit numbers.
We observe that we can place any one of the 8 digits at units place (Repetition allowed). However, 0 can’t be placed at tens place. So, we can have any one of the 7 numbers at the tens place.
Thus, the total number of 2 digit numbers which can be formed $=8\times 7=56$.
We will now consider the 3 digit numbers.
We observe that we can place any one of the 8 digits at units and tens place (Repetition allowed). However, 0 can’t be placed at hundreds place. So, we can have any one of the 7 numbers at the hundreds place.
Thus, the total number of 3 digit numbers which can be formed $=8\times 8\times 7=448$.
We will now consider the 4 digit numbers.
We observe that we can place any one of the 8 digits at units, tens and hundreds place (Repetition allowed). However, 0 can’t be placed at thousands place. So, we can have any one of the 7 numbers at the thousands place.
Thus, the total number of 4 digit numbers which can be formed $=8\times 8\times 8\times 7=3584$.
So, the total numbers less than 10000 are $=3584+448+56+8=4096$.
Hence, there are 4096 numbers less than 10000 which can be formed from the given digits.

Note: It’s necessary to keep in mind that the repetition of digits is allowed. Also, one must consider all the cases in which 4, 3, 2 and 1 digit numbers can be formed. If we don't consider all the cases, we will get an incorrect answer.