Question 1210184: I can only remember a seven-digit telephone number if the first three digits (the "prefix") are equal to the next three digits or the last three digits. For example, I can remember 389-3892 and 274-9274.
How many seven-digit telephone numbers can I remember?
(For this problem, a telephone number cannot start with a 0.)
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem into two cases based on the given condition.
**Case 1: The first three digits equal the next three digits.**
1. **First three digits:** Since the first digit cannot be 0, there are 9 choices for the first digit (1-9). For the second and third digits, there are 10 choices each (0-9).
* Number of choices for the first three digits: 9 * 10 * 10 = 900
2. **Next three digits:** These digits must be the same as the first three digits, so there's only 1 choice for each of these three digits.
* Number of choices for the next three digits: 1
3. **Last digit:** There are 10 choices for the last digit (0-9).
* Number of choices for the last digit: 10
4. **Total numbers in Case 1:** 900 * 1 * 10 = 9000
**Case 2: The first three digits equal the last three digits.**
1. **First three digits:** Same as Case 1, there are 900 choices.
* Number of choices for the first three digits: 900
2. **Next digit:** There are 10 choices for the next digit (0-9).
* Number of choices for the next digit: 10
3. **Last three digits:** These digits must be the same as the first three digits, so there's only 1 choice for each of these three digits.
* Number of choices for the last three digits: 1
4. **Total numbers in Case 2:** 900 * 10 * 1 = 9000
**Overlapping Case**
We've counted the numbers where the first three digits are equal to both the next three and the last three twice (e.g., 123-123123). We need to subtract these numbers once to avoid overcounting.
1. **First three digits:** 900 choices.
2. **Next three digits:** 1 choice.
3. **Last three digits:** 1 choice.
4. **Total numbers in the overlap:** 900 * 1 * 1 = 900
**Final Calculation**
* Total numbers = (Numbers in Case 1) + (Numbers in Case 2) - (Numbers in the overlap)
* Total numbers = 9000 + 9000 - 900 = 17100
**Therefore, you can remember 17,100 seven-digit telephone numbers.**
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
I can only remember a seven-digit telephone number if the first three digits (the "prefix")
are equal to the next three digits or the last three digits.
For example, I can remember 389-3892 and 274-9274.
How many seven-digit telephone numbers can I remember?
(For this problem, a telephone number cannot start with a 0.)
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Counting in the post by @CPhill is INCORRECT.
It becomes incorrect, when CPhill starts consider the overlapping case.
Namely, he suddenly considers then the 9-digit telephone numbers instead of 7-digit numbers.
In reality, the overlapping case may really have place.
It happens when in the form ABC-XABC part XAB is the same as ABC.
But it happens if and only if A=X, B=A, C = B, i.e. A = B = C.
The number of such cases (overlapping) is 9 (not 900, as @CPhill counts them).
Therefore, the final answer in the problem is 9000 + 9000 - 9 = 18000-9 = 17991.
ANSWER. You can remember 17991 telephone numbers.
Solved correctly.
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