Pair of numbers

Question

Suppose a “Secret Pair” number is a four-digit number in which two adjacent digits are equal and the other two digits are not equal to either one of that pair or each other. For example, 2209 and 1600 are “Secret Pair” numbers, but 1333 or 2552 are not. How many “Secret Pair” numbers are there?

(A) 720

(B) 1440

(C) 1800

(D) 1944

(E) 2160

(A) 720

(B) 1440

(C) 1800

(D) 1944

(E) 2160

ShankSouljaBoi · Answer

https://gmatclub.com/forum/suppose-a-se ... 35142.html discussed here

eswarchethu135 · Answer

There exist three types of number cases to be solved.

i) _ _ 00
ii) 11_ _
iii) _00_

In case i)

For _ _00, The last 2 digits can be from 00,11,22,...99 => 10 possibilities
The first digit can be filled in 9 ways out of 10 numbers (0 to 9).
The second digit can be filled in 8 ways out of 10 numbers, in which 0 and the number filled in first digit's place should be excluded. So total  9*8 = 72  ways

Now, consider _ _ 11, _ _ 22,...etc
The first digit can be filled in 8 ways as 0 can't be filled in the first digit and 1 also cannot be included. 
the second digit can be filled in 8 ways, as 0 can be included now but the digits in first and last position of the number should be excluded. So a total of  8*8 = 64  ways.

These 64 ways are common for _ _11 to _ _ 99. So  64*9 = 576  ways

Total possible numbers for case i) are  576+72 = 648  numbers

In case ii)

The first two digits can be from 11 to 99, 9 ways.
The third digit can be filled in 9 ways and the fourth digit can be filled in 8 ways. So total ways are  9*8 = 72  ways.

Total possible numbers are  72*9 = 648  numbers.

In case iii)

For _00_, the first digit can be filled in 9 ways and the last digit can be filled in 8 ways. Total ways are  9*8 = 72  ways
For _11_, _22_, .... _99_ , The first digit can be filled in 8 ways (excluding 0 and digit in second place) and the last digit can be filled in 8 ways (including 0 and excluding 2 other digits already in the number). So total ways are  8*8 = 64  ways for each possibility and there are 9 such possibilities, so total ways are  64*9 = 576  ways.

Total possible numbers are  576 + 72 = 648  ways.

Finally, total possible numbers are  648 +648 +648 = 1944  numbers.

OPTION :  D

Kudos, if you liked my explanation

Bunuel · Answer

Discussed here: suppose-a-secret-pair-number-is-a-four-digit-number-in-which-two-adj-235142.html Please read carefully and follow our rules of posting: rules-for-posting-please-read-this-before-posting-133935.html Thank you. THIS TOPIC IS LOCKED AND ARCHIVED.

Pair of numbers

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards