Question

An intelligence agency forms a code of two distinct digits selected from 0,1,2,3,……9 such that the first digit of the code is non-zero. The code handwritten on a slip, can however potentially create confusion when read upside down i.e, the code 91 may appear as 16.How many codes are there for which no confusion can arise.
(a) 80
(b) 78
(c) 71
(d) 60

Answer 1

Hint: From the problem, we can clearly see that code is formed by two distinct digits. So, we find a total no. of codes formed by using digits in which the first digit is non-zero. Then we find a total no. of codes which can create potential confusion when reading upside down. We can then observe that there are some codes which can be read the same even while reading upside down. So, we subtract total no. of codes which can create confusion and add the codes which can read the same upside down from total no. Of codes generated by intelligence agencies.

Complete step-by-step answer:
According to the problem, we have an Intelligence agency creating a code of two distinct digits. Here digits are taken from 0,1,2,3,……9.
We can observe that the first digit of the code is non-zero.
Let us find the total no. of codes formed by these digits.
Let us assume that the code is in the form of ‘xy’. Now, we find the total number of possibilities for both x and y.
First we find the total no. of possibilities to select digit for x.
We can see that the first digit of the problem cannot be zero as per the problem. So, we have 9 ways to select the digit for x.
So, the total no. of ways to select a digit for x is 9..
Now, we find a total no. of possibilities for y.
We can see x and y are to be distinct as per the problem. So, we cannot repeat the digit which is already selected for x. But there is no limitation on ‘0’ to select for ‘y’.
So, if we neglect the digit which is already selected for x, we remain with 9 more digits to choose one for y.
So, the total no. of ways to select a digit for y is 9.
We know that if there are ‘m’ ways to select an item for ‘x’ and ‘n’ ways to select an item for ‘y’ in order to make a number ‘xy’. The no. of ways to make a digit for ‘xy’ is ‘m.n’
So, the total no. of codes obtained by the intelligence agency is $9\times 9=81$ ways.
Now we find the no. of codes which can make confusion while reading upside down. We need to check which digits can make confusion while reading down.
Such type of confusion only of the selected digits are 1,6,7,8.
We have 4 ways to select a digit for x and 3 ways to select a digit for y as we cannot select the digit we selected for y.
We have a total of $4\times 3=12$ ways which may cause confusion.
But there are two codes 69 and 96 which look the same even while reading upside down.
Total no. of codes which doesn’t make confusion while reading upside down are $81-12+2=71$ .
∴ The total no. of codes which doesn’t make confusion are 71.
So, the correct answer is “Option C”.

Note: Here we have high chances of forgetting that 69 and 96 can read the same while reading upside down. Confusions may be that 60, 10, 80 and 90 are not read the same when they read upside down but 0 will come first when they are read upside down which is contradiction to the statement given in our problem.