Question

Let K be the number of all whole numbers formed on the screen of a calculator which can be recognised as numbers with correct digits when they are read inverted. The greatest number formed on its screen is 999999. Find the sum of digits of K?

Answer 1

Hint: We first find the possible outcomes for the inverted scenario. We find the number of possible cases which don’t have 0 in its unit place. We find the total number and then find the sum of the digits.

Complete step by step solution:
There are in total six places in the calculator screen to fill up with numbers as the greatest number formed on its screen is 999999.
Now if there is a number on the screen, then we are going to read it invertedly.
We know that a number can’t start with 0.
In inverted form, that starting number will be the last digit or the unit placed digit. The numbers which are recognizable as digits in its inverted form are $ 0,1,2,5,6,8,9 $ .
Therefore, we can’t have 0 in the unit place of the number on the calculator’s screen when we are reading it invertedly.
So, the total number of possible outcomes for one-digit numbers is 7.
Total number of possible outcomes for two-digit numbers is \[6\times 6=36\] .
Total number of possible outcomes for three-digit numbers is $ 6\times 7 \times 6=252 $ .
Total number of possible outcomes for four-digit numbers is $ 6\times 6\times {{7}^{2}}=1764 $ .
Total number of possible outcomes for five-digit numbers is $6\times 6\times {{7}^{3}}=12348 $ .
Total number of possible outcomes for six-digit numbers is $6\times 6\times {{7}^{4}}=86436 $ .
Total being $ 86436+12348+1764+252+36+7=100843 $ .
The sum of the digits will be $ 1+0+0+8+4+3=16 $ .
So, the correct answer is “ 16”.

Note: The case would have been different if the digits as input would have been unique in every input. We could not have used repetition of the numbers. The total number would have been very less than that.