Bunuel wrote:
A “palindromic integer” is an integer that remains the same when its digits are reversed. So, for example, 43334 and 516615 are both examples of palindromic integers. How many 6-digit palindromic integers are both even and greater than 400,000?
(A) 200
(B) 216
(C) 300
(D) 400
(E) 2,500
Kudos for a correct solution.
Gmatprepnow Official Solution:Can we take the task of “building” 6-digit palindromic integers and break it into individual stages? Yes!
Let’s say that our 6-digit integer takes the form XYZZYX. Once we choose values for X, Y and Z, we can be assured that the resulting 6-digit integer is palindromic.
From here, we can define the stages as follows:
Stage 1: Choose a value for X
Stage 2: Choose a value for Y
Stage 3: Choose a value for Z
Once we complete all 3 stages, we will have “built” our 6-digit palindromic integer. So, at this point, we must determine the number of ways to accomplish each stage.
Stage 1: Choose a value for X: In how many ways can we choose a value for X? Since the resulting 6-digit integer must be both even AND greater than 400,000, X must equal 4, 6 or 8. So, Stage 1 can be accomplished in 3 different ways.
Stage 2: Choose a value for Y: Since there are no restrictions on the value of Y, we can choose any of the ten digits {0,1,2,3,4,5,6,7,8,9}. So, Stage 2 can be accomplished in 10 different ways.
Stage 3: Choose a value for Z: Since there are no restrictions on the value of Z, we can choose any of the ten digits {0,1,2,3,4,5,6,7,8,9}. So, Stage 3 can be accomplished in 10 different ways.
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create an even, 6-digit palindromic number that’s greater than 400,000) in 3 × 10 × 10 ways (300 ways). So,
the answer to the original question is C.