How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if the repetition of the digits is not allowed? (a) 50 (b) 70 (c) 100 (d) 120
ANSWER
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Hint: Count all the possible three-digit numbers which can be formed using the digits 0, 1, 2, 3, 4, and 5, keeping in mind that the repetition of digits is not allowed. To count the number of possible three-digit numbers, count the possible numbers which can be placed at ones, tens, and hundreds place. Multiply all of them to get the total number of three-digit numbers.
Complete step-by-step answer:
We have to count all the numbers lying between 100 and 1000 which can be formed using the digits 0, 1, 2, 3, 4, and 5 such that the repetition of digits is not allowed. To do so, we will count all the possible three-digit numbers which can be formed using the given digits such that repetition is not allowed. We will first count the digits which can be placed at hundreds place. We can place any one of the numbers 1, 2, 3, 4, or 5 at hundreds place. We can’t place zero at hundreds place; otherwise, we will get a two-digit number which will be less than 100. Thus, the number of digits which can be placed at hundreds place is 5. We will now count the digits which can be placed at tens place. We can place one of the numbers 0, 1, 2, 3, 4, or 5 at tens place. However, we have already placed any one of the numbers from 1, 2, 3, 4, or 5 at hundreds place. Thus, we can’t place this digit at tens place as the repetition of digits is not allowed. Thus, the number of digits which can be placed at tens place is 5. We will now count the digits which can be placed at units place. We can place one of the numbers 0, 1, 2, 3, 4, or 5 at units place. However, we have already placed two of these numbers at hundreds and tens place. Thus, we can’t place those two digits at units place as the repetition of digits is not allowed. Thus, the number of digits which can be placed at units place is 4. So, the possible number of three-digit numbers is $=5\times 5\times 4=100$. Hence, we can form 100 numbers lying between 100 and 1000 using the given digits such that the repetition of digits is not allowed.
Therefore option (c) is the correct answer.
Note: One must keep in mind that we can’t place 0 at hundreds place; otherwise, we will get a two-digit number. Also, it’s important to consider that repetition of digits is not allowed. So, we can place each digit only once.