Question

The U.S. Postal Service began using five-digit zip codes in 1963. Every post office was given its own zip code, which ranged from 00601 in Adjuntas, Puerto Rico, to 99950 in Ketchikan, Alaska.

a. If the only five-digit zip code that could not be used was 00000, how many zip codes were possible in 1963?

b. Some five-digit zip codes are prone to errors because they are still legal five-digit zip codes when read upside down. When this happens, a letter goes to the wrong post office and must be returned. How many zip codes are legal when read upside down?

c. How many of the zip codes you counted in part b are not prone to errors because they read the same when turned upside down?

Answer 1

Answer:

Given Data

The U.S postal Service bagan using 5 digit zip codes

which ranged from 00601 in adjuntas , puerto Rico , to 99950 in ketchikan ,Alaska.

A)If the only five digit zip code that could not be used was 00000, how many zip codes were possible in 1963.

No . of possible zip codes = 95 - 1

because , As each of the 5 positions can be filled in 9 ways ,

i.e with 9 possible numbers and 1 is subtracted as 00000 not allowed.

b)Some 5 digit zip codes are prone to errors because they are still legal 5 digit zip codes when read upside down.

0,1, and 8 remains unchanged when upside down.

6 becomes 9 and vice versa when viewed upside down.

Number of zip codes prone to error = 55 - 1

( 1 is subtracted as 00000 is not allowed )

c) How many of the zip codes you counted in part b are not prone to errors because they read the same when turned upside down.

  

Now , clearly 6 or 9 cannot be placed in the middle most blank.

As then the zip code will not remain same , so the number of ways of filling the middle blank = 3.

Now , b and d mut simultaneously will be 1 , 8 , 0 or ( 6 , 9)

That is ; b d

1 1

   8 8

      0 0

    6 6

9 9

it is equivalent to filling b or d blank in 5 ways.

similar argument for a and c

Number of zip codes that read same when turned upside down = ( 5 * 5 * 3 ) -1

= 75 -1

= 74

[as 00000 is not allowed .]