Problem 1 Part b – Correct! – Score: 3 / 7 (1923)
Report an ErrorProblem:
We call a number special if every digit in the number either is a 1 or borders a 1. For example, 11111, 13, 141, 1441, 515151, and 101 are all special, but 10001, 222, 122, and 1333 are not special.
How many positive 3-digit numbers are special?
How many positive 3-digit numbers are special?
Solution:
A 3-digit number is special if it has a 1 in the middle, or it both starts and ends in a 1. There are 
of the first type (we have 9 choices for the first digit and 10 choices for the last digit). There are also 10 of the second type (we must choose the middle digit), but we already counted 111 in the first case, so we don't count it again; we get 9 new special numbers in the second case. Therefore, there are 90 + 9 = 99 special three-digit numbers.
of the first type (we have 9 choices for the first digit and 10 choices for the last digit). There are also 10 of the second type (we must choose the middle digit), but we already counted 111 in the first case, so we don't count it again; we get 9 new special numbers in the second case. Therefore, there are 90 + 9 = 99 special three-digit numbers.
Your Response(s):
- :( 81
- :( 82
- :) 99
Analysis: The formats 1_1, _1_, and 111 aren't enough - the numbers can be in the form of 11_ or _11 too!
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