Of the four-digit integers greater than 9000

Question

Of the four-digit integers greater than 9,000, with the sum of their digits greater than 30, what is the sum of the integers that have two pairs of digits such that digits in one pair are equal to each other and the digits of the other pair are equal to each other but distinct from the other two?

A. 0
B. 9,999
C. 59,328
D. 88,659
E. 270,000

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A. 0
B. 9,999
C. 59,328
D. 88,659
E. 270,000

MathRevolution · Accepted Answer

Four-digit integers greater than 9,000 -> This means one of the number has to be 9.(The first digit )
Digits in one pair are equal to each other --> This means another number has to be 9 as well.

First Possible combinations:  99_ _, 9_ 9_ and 9 _ _ 9.

Sum of their digits greater than 30: Two numbers are 9 and 9. So 30-18 = 12 . The sum of remaining two digits should be greater than 12 and and the digits of the other pair are equal -> This means the both the numbers will be same.

Possible combination:  7 + 7 = 14 and 8 + 8 = 16

Placing the digit (7) in first possible combination: 9977, 9797, 9779.
Placing the digit (8) in first possible combination: 9988, 9898, 9889.

Adding all 6 results: 9977 + 9797 + 9779 + 9988 + 9898 + 9889 = 59,328.

Answer C

kush09 · Answer

Of the four-digit integers greater than 9,000, with the sum of their digits greater than 30, what is the sum of the integers that have two pairs of digits such that digits in one pair are equal to each other and the digits of the other pair are equal to each other but distinct from the other two?

A. 0
B. 9,999
C. 59,328
D. 88,659
E. 270,000

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Solution:  
four-digit integers greater than 9,000:
 9001 to 9999

sum of their digits greater than 30:
 Only possible 4 digits are 6,7,8,9

one pair are equal to each other and the digits of the other pair are equal to each other but distinct, 
 means of 4 digits, 2 different integers have to be selected, that gives us 2 set (9,6) and (8,7)

as total 12 different integers are possible :
9669,9966,9696,6699,6969,6996,8778,8787,8877,7887,,7878,7788

Approximately, answer is 12*9000 = 10800, answer should be greater than 10,800,  which is only option E .

Could you please share the solution, how OA is C?

Gknight5603 · Answer

9669,9966,9696,6699,6969,6996,8778,8787,8877,7887,,7878,7788

these are not the desired solutions....

only 9977,9988,9779,9889,9898,9797. are the desired solutions.....
numbers should be greater than 9000 and sum of their digits should be greater than 30

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saurabh851 · Answer

Ans: C

Solution:
four-digit integers greater than 9,000
  9001 - 9999

Sum of their digits greater than 30,
  Possible values are 9, 7, 8 only

One pair are equal to each other and the digits of the other pair are equal to each other but distinct from the other two,
Possible four digit integers satisfied the above condition are-
  9977, 9988 only

Total 6 different arrangements are possible-
  9977, 9779, 9797, 9988, 9889, 9898

Total sum: 9977 + 9779 + 9797 + 9988 + 9889 + 9898 = 59,328

saurabh851 · Answer

Your solution doesn't satisfy the conditions mentioned in the question as-
1. Four-digit integers  greater than 9,000 
2. Sum of their digits  greater than 30

kush09 · Answer

Thank you, I miss read sum should be  greater that 30 . I calculated solution to sum is equal to 30.

rsrighosh · Answer

Why are we considering 9797, 9779, 9898, 9889

As per my understanding these numbers do not satisfy 
 that have two pairs of digits such that digits in one pair are equal to each other and the digits of the other pair are equal to each other but distinct from the other two?

or is it like we have to find all combinations with such digits?

I was initially going by the numbers 9977 and 9988 only

StacyArko · Answer

Thanks so much. How about 7799, 7979, 7997, 8899, 8989, 8998. Do these 6 options also qualify? Thanks in advance.

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Bunuel · Answer

No. Check the question:

Of the four-digit integers  greater than 9,000 , ...

pappal · Answer

since the no. is greater than 9000 and a four digit integer so one of the pair must be 99 also first digit must be 9.
and the sum of the digits is greater than 30 so the only other pairs are 77 or 88.
now its to find out how many different no.'s can be formed? and their sum?
9988,9898 & 9889
9977,9797 & 9779 are only possible numbers.
since sum of their units digits is 48 hence the  correct choice is C  (no need to calculate the complete summation)

Of the four-digit integers greater than 9000

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Of the four-digit integers greater than 9000

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