three-digit

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three-digit

by GmatKiss » Sat May 26, 2012 1:48 pm
Of the three-digit integers greater than 660, how many have two digits that are equal to each other and the remaining digit different from the other two?
(A) 47
(B) 60
(C) 92
(D) 95
(E) 96
Source: — Problem Solving |

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by digvijayk » Sat May 26, 2012 2:56 pm
IMO: D

Solution:
Generally, if we say 10 to 90, we use the formula-> (90-10)+1= 81, to determine the number of numbers.
But, here, it will be simpler since its written that the numbers shoul be greater than 660.
So, the total number of numbers in this set = 999-660 = 339.

We know that 4 numbers are such which have all three digits same => Count=4(666,777,888,999)
We can find out the number of digits with distinct digits= (1x3x8) + (3x9x8)= 240, Count = 240
So the ones left are the ones we're looking for => 339-4-240=95!
So, D.

Brute force:
661,662,663,664,665,667,668,669,676,677,686,688,696,699, 700
707,711,717,722,727,733,737,744,747,755,757,766,767,770,771,772,773,774,775,776,778,779,787,788,797,799, 800
808,811,818,822,828,833,838,844,848,855,858,866,868,877,878,880,881,882,883,884,885,886,887,889,898,899, 900
909,911,919,922,929,933,939,944,949,955,959,966,969,977,979,988,989,990,991,992,993,994,995,996,997,998