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26 May 2018, 07:55
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
19% (02:32) correct
81%
(02:22)
wrong
based on 220
sessions
History
Date
Time
Result
Not Attempted Yet
How many positive seven-digit integers include the sequence 123, in that order? (For example, 1234567 and 9991239)
A. 45,971
B. 46,000
C. 47,979
D. 49,961
E. 50,000
A. 45,971
B. 46,000
C. 47,979
D. 49,961
E. 50,000
26 May 2018, 08:22
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Bunuel
Below are the cases to consider :
a.123 _ _ _ _ : each place can be occupied by integers 0,1,2,....9 i.e., 10 ways. Hence total ways=\(10^4\)
b._ 123 _ _ _ : 1st place can be filled up with 9 ways (1,2,...9)and other 3 each in 10 ways similar to case a. Hence total ways=\(9*10^3\)
c._ _123 _ _ : Hence total ways=\(9*10^3\)
d._ _ _ 123 _ : Hence total ways=\(9*10^3\)
e._ _ _ _ 123 : Hence total ways=\(9*10^3\)
Now, there are some cases of double counting in the above:
A. 123_123: 10 ways
B.123123_ : 10 ways
C._123123 : 9 ways
Hence total unique ways = \((a+b+b+d+e)-(A+B+C)\)= \(10^4\)+\(9*10^3\)+\(9*10^3\)+\(9*10^3\)+\(9*10^3\) \(-(10+10+9)=46000-29=45971\)
Hence I will go for option A.
General Discussion
26 May 2018, 08:34
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Lets take "123" as one entity. All Possible combination are:
1. 123_ _ _ _ i.e for remaining places can be filled by 10*10*10*10 => 10^4 ways
2. _ 123 _ _ _ => 9 * 10*10*10 => 9*10^3 ways.
3. _ _ 123 _ _ => 9 * 10*10*10 => 9*10^3 ways.
4. _ _ _ 123 _ => 9 * 10*10*10 => 9*10^3 ways.
5. _ _ _ _ 123 => 9 * 10*10*10 => 9*10^3 ways.
Adding all => 10^4+ 4*9*10^3 => 10^3 (10+36) => 46*10^3 ways i.e. 46,000
Now, there are some cases of double counting in the above:
1. 123_123: 10 ways
2. 123123_ : 10 ways
3. _123123 : 9 ways
i.e total of 29 double counting.
Hence total unique possibilities are 46000-29 = 45,971.
A is the answer.
1. 123_ _ _ _ i.e for remaining places can be filled by 10*10*10*10 => 10^4 ways
2. _ 123 _ _ _ => 9 * 10*10*10 => 9*10^3 ways.
3. _ _ 123 _ _ => 9 * 10*10*10 => 9*10^3 ways.
4. _ _ _ 123 _ => 9 * 10*10*10 => 9*10^3 ways.
5. _ _ _ _ 123 => 9 * 10*10*10 => 9*10^3 ways.
Adding all => 10^4+ 4*9*10^3 => 10^3 (10+36) => 46*10^3 ways i.e. 46,000
Now, there are some cases of double counting in the above:
1. 123_123: 10 ways
2. 123123_ : 10 ways
3. _123123 : 9 ways
i.e total of 29 double counting.
Hence total unique possibilities are 46000-29 = 45,971.
A is the answer.
Glad it added some value 
