There are 900 3-digit numbers, for we can choose the first digit
9 ways, the 2nd digit 10 ways, and the 3rd digit 10 ways.
So we will subtract from 900 all the 3-digit numbers that
have at least one pair of consecutive digits.
Let A = the set of all 3 digit numbers with the first two digits
consecutive.
Let B = the set of all 3 digit numbers with the second two digits
consecutive.
Then we will use
N(A or B) = N(A) + N(B) - N(A and B)
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N(A) =
N(3-digit integers with the first two digits consecutive)
1. There are 8 ways to choose the first digit as 1 through 8,
There are then 2 ways to choose the second digit consecutive with the
first digit.
There are then 10 ways to choose the third digit.
That's 8x2x10 = 160
2. There is 1 way to choose the first digit as 9,
There is then 1 way to choose the second digit consecutive with the
first digit, namely to choose it as 8
There are then 10 ways to choose the third digit.
That's 1x1x10 = 10
So N(A) = 160+10 = 170 ways,
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N(B) =
N(3-digit integers with the last two digits consecutive)
1. There are 8 ways to choose the last digit as 1 through 8,
There are then 2 ways to choose the second digit consecutive with the
last digit.
There are then 9 ways to choose the first digit.
That's 8x2x9 = 144
2. There are 2 ways to choose the last digit as 0 or 9,
There is then 1 way to choose the second digit consecutive with the
last digit
There are then 9 ways to choose the first digit.
That's 2x1x9 = 18
Or a total of 144+18 = 162 ways,
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N(A and B)
N(3-digit integers with both the first two and the last two digits
consecutive) =
1. There are 6 ways to choose the first digit 2 through 7
There are then 2 ways to choose the second digit consecutive with
the first digit. There are then 2 ways to choose the third digit
consecutive with the second digit.
That's 6x2x2 = 24 ways.
2. There are 3 ways to choose the first digit 1, namely 101, 110, and 121
3. There are 3 ways to choose the first digit 8, namely 878, 879, and 898
4. There are 2 ways to choose the first digit 9, namely 987 and 989.
That's 8 ways.
So N(A and B = 24+8 = 32
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Therefore N(A or B) = N(A) + N(B) - N(A and B) = 170 + 162 - 32 = 300,
Therefore the answer to the problem is 900 - 300 or 600.
Edwin
