Write the 7−digits largest and smallest number having three different digits and also find their sum.

Hint- We will follow a simple approach to solve this question since we are given that both the numbers must contain three different digits so in case of 7-digits largest number we will use the digit 9, 8, 7 in such a way that all the digits will be 9 except the last two will be 8, 7 respectively to form the required 7−digits largest number having three different digits and in case of 7-digits smallest number we will use the digit 1, 0, 2 in such a way that all the digits will be 0 except the most significant digit will be 1 and least significant digit will be 2 to form the required 7−digits smallest number having three different digits. In the end, we will just find their sum.

Complete step-by-step solution -
Let us first consider the case for 7 – digits largest number having three different digits
Let the largest 7 – digit number begin with as many nines as possible but since there must be three different digits so let us begin with five nines and fill the last two places with 8 and 7 respectively.
So, the largest 7 – digit number is 9999987
Let us first consider the case for 7 – digits smallest number having three different digits
Since, the smallest 7 – digit number is 1000000 but since there must be three different digits so let us change the least significant digit with 2.
So, the smallest 7 – digit number is 1000002
Now their sum $ = 9999987 + 1000002 = 10999989$
Hence, the sum of the 7−digits largest and smallest numbers having three different digits is 10999989.

Note- For such types of questions, just keep in mind that when forming the largest number with 3 different digits, focus on digits 9, 8, 7 and when forming the smallest number with 3 different digits, focus on digits 0, 1, 2 and rearrange them according to the condition given. Sometimes students consider the leftmost digit as 1 which is wrong because here we have to choose 3 different digits. We have already chosen two digits 0,1 so the least remaining digit is only 2.