5lb Book, 2nd Edition, Chapter 30, Advanced Quant, Ques.19

[prashantnigam16]

Q. How many positive 5 digit integers contain the digit grouping "57" (in that order) at least once? For instance 30457 and 20574 are two such integers to include, but 30475 and 20754 do not meet the restrictions.

A) 279
B) 3000
C) 3500
D) 3700
E) 4000

It's answer should be 3671 and not 3700 (option D).

And here's what the solution in the book precisely says:
Cases <---------------> Possible Numbers
1. 57 _ _ _ <---------------> 1000
2. _ 57 _ _ <---------------> 900
3. _ _ 57 _ <---------------> 900
4. _ _ _ 57 <---------------> 900
___________________________________
Total = 3700

Now, this is what I think,

Cases 1&3 have 10 common numbers being 57570, 57571, ... , 57579
Cases 1&4 have 10 common numbers being 57057, 57157, ... , 57957
Cases 2&4 have 9 common numbers being 15757, 25757, ... , 95757
These sum up for 29 common numbers counted twice in 3700, so must be removed once, making the total possible numbers to 3700-29=3671
Therefore 3671 should be the correct answer and not 3700.
If my solution is correct then please acknowledge.

And thanks to this book, I scored a full quant score of 170 in my GRE recently!

[tommywallach]

Hey P,

Unfortunately, you are incorrect. The duplication is irrelevant, because the question simply asked "How many have 57 AT LEAST ONCE." Duplicates have not been counted in the correct answer, nor should they be.

57_ _ _

This grouping features 1000 numbers: everything from 57,000 to 57,999 (i.e. everything from 000 to 999). The numbers 57,570 and 57,057 have not been counted twice, nor should they be.

_ 57 _ _

This grouping features 900 numbers. It's not 1000 because we can't have 0 for the first digit. The 900 numbers would be everything from 15,700 to 95,799 (i.e. everything from 100 to 999).

There is nothing that should be subtracted from the total reached this way.

-t

[prashantnigam16]

Hey Tommy,

I understand the solution given in the book and which you reiterated. And I also understand what "How many have 57 AT LEAST ONCE." means, it means 15700 (57 once) and 15757 (57 twice) both qualify. You say duplicates have not been counted, so can you disprove the following:

To take up the following 2 patterns - 57 _ _ _ and _ _ 57 _

57 _ _ _ includes 57000 to 57999, including { 57570, 57571, 57572, 57573, 57574, 57575, 57576,57577, 57578, 57579 }. Thus 10*10*10 = 1000 numbers
and
_ _ 57 _ also includes { 57570, 57571, 57572, 57573, 57574, 57575, 57576,57577, 57578, 57579 } because the first digit can be anything but 0 and the second and last digit can be 0-9. So the first digit can be 5 (as 1-9 are allowed), the second digit can be 7 (as 0-9 are allowed) and the last digit can be any of 0-9. So this pattern is capable of generating 5757 _ as is the first pattern. Also, this pattern has a total of 9*10*10 = 900 numbers.

The book's solution adds up 1000 and 900 making the count 1900. What it does is that it counts { 57570, 57571, 57572, 57573, 57574, 57575, 57576,57577, 57578, 57579 } numbers once for 57 _ _ _ and then once again for _ _ 57 _ , thus creating duplicates.

The numbers in this set are clearly being counted twice { 57570, 57571, 57572, 57573, 57574, 57575, 57576,57577, 57578, 57579}, hence the count of numbers need to be reduced by 10 as there are 10 such numbers.

Similarly the numbers in these 2 sets {57057, 57157, 57257, 57357,57457,57575,57657,57757,57857,57957} and {15757, 25757, 35757, 45757, 55757, 65757, 75757, 85757, 95757} are also being counted twice.

I further cross-checked my count with a computer program and the count is actually 3671 and not 3700. 3700 is actually the number of times 57 occurs in all 5 digit natural numbers, which is clearly not what the question had asked. By number of times I mean like in 15757 "57" occurs twice and in 15700 it occurs once, and thus counting the number of all such occurences.

[tommywallach]

You are entirely right! My apologies. I misread the original question, so then I didn't even understand your explanation. My apologies! We'll get this fixed up!

-t