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[Martina1120]

Hi I am working on a problem as follows:

Students log on to the California Virtual Campus with a user name consisting of eight characters: four uppercase letters of the alphabet followed by four digits.

a. How many user names are theoretically possible for this system?
26*26*26*26*10*10*10*10= 4,569,760,000

b. How many user names have no repeated letters or digits?
26*25*24*23*10*9*8*7=1,808,352,000

this is where i got stuck:

c. How many user names have no matching adjacent letters or digits?I dont see the difference between this and question b? the solution is : 26*25*25*25*10*9*9*9=2,961,562,500 but why is just the second option repeated?

d. How many user names begin with A and end with 9? I don't even know where to start with this one.

Any help that anyone can provide would be greatly appreciated. I know this is simple stuff, but I'm just not catching on right away. Thanks!

 

[Deleted member 4993]

Hi I am working on a problem as follows:

Students log on to the California Virtual Campus with a user name consisting of eight characters: four uppercase letters of the alphabet followed by four digits.

a. How many user names are theoretically possible for this system?
26*26*26*26*10*10*10*10= 4,569,760,000

b. How many user names have no repeated letters or digits?
26*25*24*23*10*9*8*7=1,808,352,000

this is where i got stuck:

c. How many user names have no matching adjacent letters or digits?I dont see the difference between this and question b? the solution is : 26*25*25*25*10*9*9*9=2,961,562,500 but why is just the second option repeated?

Say you choose A for first letter (26 choices) - then the next letter cannot be A (25 choices - say you choose B) - now for the third letter A is free to be chosen (that was not the case in (b)) and you have 25 choices......
d. How many user names begin with A and end with 9? I don't even know where to start with this one.

For the first choice you have only one choice - 'A' and for the last space you have only one choice '9'. Rest is like (a)
Any help that anyone can provide would be greatly appreciated. I know this is simple stuff, but I'm just not catching on right away. Thanks!

 

[soroban]

Hello, "Martina1120!

Students have a user name consisting: four uppercase letters of the alphabet followed by four digits.

c. How many user names have no matching adjacent letters or digits?

The 1st character can be any letter: 26 choices.
The 2nd must not match the 1st one: 25 choices.
The 3rd must not match the 2nd one: 25 choices.
The 4th must not match the 3rd one: 25 choices.

The 5th can be any digit: 10 choices.
The 6th must not match the 5th: 9 choices.
The 7th must not match the 6th: 9 choices.
The 8th must not match the 7th: 9 choices.

Therefore, there are: .\(\displaystyle 26\cdot25\cdot25\cdot25\cdot10\cdot 9\cdot9\cdot 9\) ways.

d. How many user names begin with A and end with 9?

The first character is \(\displaystyle A\) and the last is \(\displaystyle 9.\)

The user name looks like this: .\(\displaystyle A\:\_\;\_\;\_\;\_\;\_\;\_\;9\)
. . and we want three letters followed by three digits in the spaces.

There are: .\(\displaystyle 26\cdot26\cdot26\cdot10\cdot10\cdot10\) ways.

 

[Martina1120]

Thanks so much to both posters! I will get the hang of this eventually