How Do Restrictions Affect Counting in Combinatorics?

[n00neimp0rtnt]

Homework Statement

1. How many strings of eight English letters are there if no letter can be repeated?
2. How many strings of eight English letters are there if X is the first letter and no letter can be repeated?
3. How many strings of three decimal digits do not contain the same digit three times?

The Attempt at a Solution

You're probably able to tell that all of these problems follow a similar pattern: restrictions on how often a letter/number can show up. Other problems on the same assignment ask WITHOUT restriction (How many strings of eight English letters are there if no letter can be repeated?) and can therefore be solved with an easy exponent (26⁸), but I'm not sure how to do these ones. I do believe, however, they somehow involve a factorial, as I recall examples in class demonstrating use of them.

Possibly something like 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19, since after you pick one of 26, there are only 25 left to choose, and after 25 there are only 24 left to choose, and so on.

Thanks in advance!

 

[Mark44]

Homework Statement

1. How many strings of eight English letters are there if no letter can be repeated?
2. How many strings of eight English letters are there if X is the first letter and no letter can be repeated?
3. How many strings of three decimal digits do not contain the same digit three times?

The Attempt at a Solution

You're probably able to tell that all of these problems follow a similar pattern: restrictions on how often a letter/number can show up. Other problems on the same assignment ask WITHOUT restriction (How many strings of eight English letters are there if no letter can be repeated?) and can therefore be solved with an easy exponent (26⁸), but I'm not sure how to do these ones. I do believe, however, they somehow involve a factorial, as I recall examples in class demonstrating use of them.

Possibly something like 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19, since after you pick one of 26, there are only 25 left to choose, and after 25 there are only 24 left to choose, and so on.

Thanks in advance!

Yes, that's right for #1, assuming that all the letters are the same case, either upper case or lower case.

For #2, your thinking should be similar, but now you have only 7 letters that can vary.

For #3, it's probably easier to consider all of the possible 3-digit combinations, minus the combinations that use the same digit three times.