# Thread: More Fundamental Counting

1. ## More Fundamental Counting

How many seven-letter words contain at least one X?

How many seven-letter words contain at least two X's?

2. Originally Posted by kturf
How many seven-letter words contain at least one X?

How many seven-letter words contain at least two X's?
Once again, what. do. you. think?

3. ## Here's what I think...

At least one X:
26!/6!(26-6)! + 25!/(25-5)! + .... 22!/(22-2)! + 1

At least 2 X's would begin at 25...

Is that right? Or is it something to do with 26^7 minus something?

4. Originally Posted by kturf
At least one X:
is it something to do with 26^7 minus something?
Hint: There are $25^7$ that contain no X.

5. So would it be 25^6 that contain 1 X and 25^5 that contain 2 X's?

6. Originally Posted by kturf
So would it be 25^6 that contain 1 X and 25^5 that contain 2 X's?
Absolutely not.
Contains exactly one X: $7\cdot 25^6$.

In order to receive further help, you must reply with an explanation of why that is the correct answer.

7. For a word with 7 letters and exactly one X, the answer is 7 times 25^6 because you have seven spots available and 25 other possible letters for the 6 spots that are not the X.

So, for one or more X's, the answer would be

7 times 25^6 + 7 times 25^5 + 7 times 25^4 + 7 times 25^3 + 7 times 25^2 + 7 times 25 + 1?

8. Originally Posted by kturf
How many seven-letter words contain at least two X's?
$26^7-25^7-7\cdot 25^6$ WHY?

9. If you could explain why, that would be very helpful.

10. Originally Posted by kturf
If you could explain why, that would be very helpful.
$26^7$ the total.
$25^7$ contains no X.
$7\cdot 25^6$ contains exactly one X.
$26^7-25^7-7\cdot 25^6$ remove those two from the total.
What do you have left?