1. Consider the two dierent methods for computing the number of 4-digit pins (wh

Question

1. Consider the two dierent methods for computing the number of 4-digit pins (where each digit may either be a number (0-9) or a letter (a-z)) where at least 1 digit must be an 8.

Method 1: There are 4 positions in which the 8 can go hence we can rst choose where the 8 goes. (there are 4 such choices). Then we can choose the other digits to be any digit (there are 36 possibilities for each of these three digits). Hence the total number of such 4 digits pins is 4·36·36·36 = 186624.

Method 2: There are 36^4 total possible pins (which may or may not contain a 8), and the number of pins which do not contain an 8 is 35^4. Thus the total number of pins that contain an 8 can be derived through the dierence rule as 36^4 35^4 = 178991.
Since we have dierent results only one (or perhaps neither) of these results is correct. Which method gives the correct number of such pins and explain what is wrong with the other method.

Explanation / Answer

Method 1: This is not true. This is explained by the following example.

Suppose we initially put an 8 at the first character so we are now left with

8 _ _ _

Now suppose in choosing the second character we put 8 in the second character and then we choose a and b for the last 2 so we get:

8 8 a b

Now doing another iteration of this method. Suppose we this time put an 8 at the second place. and therefore we get:

_ 8 _ _

Now suppose while choosing the first character we choose 8 and then a and b as the last 2 characters so we again get:

8 8 a b

Therefore we see that in 2 iterations of the same method, the same combination is repeated. Therefore this method is wrong.

Method 2:

This method is true.

The total number of ways to fill each character is 364 as there are 36 ways to fill each character ( without any condition ) .

Now with the condition that there should not be any 8, we are left with 35 ways to fill each of the 4 characters. Therefore total number of combinations here would be : 354

Therefore now total number of combinations with at least one 8 would be:

= Total number of combinations ( without any condition ) - Total number of combinations such that there is no 8

= 364 - 354 = 178991

Therefore the total number of ways here are 178991

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