Question

For a 3 digit code with distinct numbers. (0-9)

How many combinations to get the code (max)?

How many combinations if you remember the middle number is 1?

Answer 1

A 3 digit code is formed with distinct numbers, from 0 to 9.

Now, from 0 to 9, there are 10 digits.

So, the first digit of the code can have 10 digits, anything from 0 to 9; so the first digit can be written in 10 ways.

Now, for any of these choices, the second digit can have 9 digits; so the second digit can be written in 9 ways, the one option left out is the digit in the first place.

Again, for any of these choices, the third digit can have 8 digits; so the third digit can be written in 8 ways, the two options left out are the two digits in the first two places.

So, by multiplication rule of combination, the code of three digits can be formed in 10*9*8, ie. 720 number of ways.

So, all possible cases is 720.

(a) So, there are maximum 720 combinations to get the code.

(b) Now, to find the number of combinations, if the middle digit is 1.

The middle digit is 1, and can be so, in only 1 way.

Now, the first digit can have 9 options then, as it cannot be 1 now. So, the first digit can be written in 9 ways.

For any of these, the third digit can have 8 options; So, the third digit can be written in 8 ways, the two options left out being 1 and the first digit.

So, by multiplication rule of combination, the code can be written in 9*1*8, ie. 72 number of ways.

So, if the middle number is 1, the maximum number of combinations is 72.