 QUESTION

# How many numbers between 10 and 10,000 can be formed by using digits 1,2,3,4,5 if digits can be repeated?

It is given in the question that we have to form numbers between 10 and 10,000 using digits 1,2,3,4,5 and when repetition is allowed. So, we will look over each fragment individually, like we cannot make numbers less than 10 and also, we cannot make numbers greater than 10,000. So, the number of 2 digit numbers that can be formed using 1,2,3,4,5 is given by $5\times 5=25$ numbers, because we can fill tens place in 5 ways and ones place of 2 digit numbers also in 5 ways.
Number of three digit numbers that can be formed using 1,2,3,4,5 is given by $5\times 5\times 5=125$ numbers, because we can fill hundreds, tens and ones place of a 3 digit number in 5 ways each. Similarly, we can fill thousands, hundreds, tens and ones place of a 4 digit number each in 5 ways. Therefore, numbers of 4 digit numbers that can be formed using digits 1,2,3,4,5 are given by $5\times 5\times 5\times 5=625$ numbers. Since, in the question, the upper limit is 10,000, so we cannot form 5 digit numbers. Therefore, the total number of digits that can be formed using digits 1,2,3,4,5 between 10 and 10,000 are given by $25+125+625=775$ .
Note: When repetition is allowed we get 775 different numbers and when repetition is not allowed we get 2 digit numbers as $5\times 4=20$ as we cannot repeat the digit at one place. Similarly, 3 digit numbers as $5\times 4\times 3=60$ and 4 digit numbers as $5\times 4\times 3\times 2=120$ . Therefore, a total of $20+60+120=200$ numbers.