Problem Solving

Dear Pros,

Pls tell which part of my calculation is wrong in answering the following question:

Of the three digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

90
82
80
45
36

Here is how i approached the problem:

we will have three cases
s s d
s d s
d s s where s represents same and d represents different

for the first case (s s d) we start with the most restrictive digit, the first one. we will have 3 options for this digit, either 7, 8, or 9

as for the third digit, which is restrictive also and must be different from the first, we can't use 0 as the number must be greater than 700 and we can't use the number in the first digit, so we have 8 options. and for the middle digit we have only 1 option for it to be similar to the last digit. so for the first case we have 3x1x8 = 24 ways

for the second case (s d s) first digit will have 3 options, last has 1 option and the middle has 9 options, so we have 3x9x1 = 27 ways

for the last case (d s s ) first digit will have 3 options, last will have 8 as it must not include a zero nor the number used in the first digit, and the middle digit will have one option giving us 3x1x8 = 24

adding the three cases we get 24+27+24 = 75 different ways!

this answer is not even in the answer choices so I must have done something wrong.

pls enlighten me

Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

Integers with exactly 2 digits the same = Total integers - Integers with all 3 digits the same - Integers with all 3 digits different.

Total integers:
To count consecutive integers, use the following formula:
Number of integers = biggest - smallest + 1.
Thus:
Total = 999 - 701 + 1 = 299.

Integers with all 3 digits the same:
777, 888, 999.
Number of options = 3.

Integers with all 3 digits different:
Number of options for the hundreds digit = 3. (7, 8, or 9)
Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.)
Number of options for the units digit = 8. (Any digit 0-9 other than the two digits already used.)
To combine these options, we multiply:
3*9*8 = 216.

Thus:
Integers with exactly 2 digits the same = 299-3-216 = 80.

The correct answer is C.

Amrabdelnaby wrote:Here is how i approached the problem:

we will have three cases
s s d
s d s
d s s where s represents same and d represents different

for the first case (s s d) we start with the most restrictive digit, the first one. we will have 3 options for this digit, either 7, 8, or 9

as for the third digit, which is restrictive also and must be different from the first, we can't use 0 as the number must be greater than 700 and we can't use the number in the first digit, so we have 8 options. and for the middle digit we have only 1 option for it to be similar to the last digit. so for the first case we have 3x1x8 = 24 ways

Case 1: SSD
The portion in red undercounts the number of options for D, since 700 is possible only in Case 3 -- DSS -- where the last two digits are the same.
In Case 1, D=0 is a valid option, yielding a total of 9 options for the value of D.
Number of ways to form SSD:
3*1*9 = 27.

for the second case (s d s) first digit will have 3 options, last has 1 option and the middle has 9 options, so we have 3x9x1 = 27 ways

for the last case (d s s ) first digit will have 3 options, last will have 8 as it must not include a zero nor the number used in the first digit, and the middle digit will have one option giving us 3x1x8 = 24

Case 3: SDD
The calculation for Case 3 omits two valid integers in which D=0:
800 and 900.
Adding these two options to the total, we get:
24+2 = 26.

Total ways = Case 1 + Case 2 + Case 3 = 27 + 27 + 26 = 80.

You might also think of it this way.

Suppose that we have our three cases: SSD, SDS, DSS. The first two cases are identical: 3 choices for S, 9 choices for D (since D can be anything other than S), so 3*9 + 3*9.

The third case "should" be identical, giving us another 3*9. But 700 isn't an option! So we only have 3*9 - 1, or 26 cases for DSS, giving us a total of 3*9 + 3*9 + 3*9 - 1, or 80.