Problem 22 · 1985 AJHSME
Stretch
Counting & Probability
restricted-count-ratio
Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1. What fraction of telephone numbers begin with 9 and end with 0?
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Answer: B — 1⁄80.
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Hint 1 of 2
A 'what fraction' question is (favorable count) ⁄ (total count). Build both counts digit-by-digit, position by position. The first digit is special (it can't be 0 or 1), and the favorable case also pins the first and last digits.
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Hint 2 of 2
Here's the shortcut: the five middle digits are totally free in BOTH counts, so they contribute the same factor on top and bottom — they cancel. Only the constrained positions (first and last digit) actually decide the fraction.
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Approach: count and take the ratio
- Total valid numbers: first digit has 8 choices (2 through 9, since 0 and 1 are banned), the other six digits have 10 each → 8 · 10⁶.
- Favorable (start 9, end 0): first digit forced to 9 (1 way), last digit forced to 0 (1 way), five middle digits free (10⁵).
- Ratio = 10⁵ ⁄ (8 · 10⁶) = 1⁄80.
- Why this transfers: in a fraction of counts, any position with the same freedom on top and bottom cancels — so you can think of just the constrained spots: 1-in-8 for the first digit being 9, times 1-in-10 for the last being 0, gives 1⁄80 directly.
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