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Bunuel
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Woah! You make it look so easy! Definite Kudos!
anuj04
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.
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Woah! You make it look so easy! Definite Kudos!
anuj04
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.
:please Glad it added some value
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Another possibly quick way is:
1) Total number of arrangement possible: 123 as a whole takes up one of 5 possible slots in the sequence of a 7-digit number and each of the other 4 slots can take one of 10 possible values (0-9). That is 5*10^4.
2) A 7-digit number cannot have it's first digit as 0. So the arrangements with first digit being 0 have 4*10^3 possibilities (i.e. 123 as a whole takes up one of 4 possible slots, each of the other slots can take 10 possible values.
3) Possibilities of 123 repetition: 123_123 (10 possibilities) or 123123_ (10 possibilities) or _123123 (9 possibilities since we have already removed all possibilities of first digit being 0 from step 2 above). That is, 29 possibilities.

Outcome: 50,000-4,000-29=45,971
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A very stupid doubt, maybe, but why do we exclude double-counting? They are also 7-digit numbers with 123.
JuniqueLid
Another possibly quick way is:
1) Total number of arrangement possible: 123 as a whole takes up one of 5 possible slots in the sequence of a 7-digit number and each of the other 4 slots can take one of 10 possible values (0-9). That is 5*10^4.
2) A 7-digit number cannot have it's first digit as 0. So the arrangements with first digit being 0 have 4*10^3 possibilities (i.e. 123 as a whole takes up one of 4 possible slots, each of the other slots can take 10 possible values.
3) Possibilities of 123 repetition: 123_123 (10 possibilities) or 123123_ (10 possibilities) or _123123 (9 possibilities since we have already removed all possibilities of first digit being 0 from step 2 above). That is, 29 possibilities.

Outcome: 50,000-4,000-29=45,971
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zaltinator
A very stupid doubt, maybe, but why do we exclude double-counting? They are also 7-digit numbers with 123.


A number like 1231234 contains two instances of 123, but it is still only one number. We are counting how many 7-digit numbers contain 123, not how many total occurrences of 123 appear, so 1231234 must be counted once, not twice.
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I missed double counting
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Having an option of 45,970, would have made it even more trickier IMO, to catch people who subtract additional cases with 0 at the start too.
Bunuel
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
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