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# How many positive seven-digit integers include the sequence 123, in th

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Re: How many positive seven-digit integers include the sequence 123, in th [#permalink]
Woah! You make it look so easy! Definite Kudos!
anuj04 wrote:
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.

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Re: How many positive seven-digit integers include the sequence 123, in th [#permalink]
abhinav770 wrote:
Woah! You make it look so easy! Definite Kudos!
anuj04 wrote:
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.

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Re: How many positive seven-digit integers include the sequence 123, in th [#permalink]
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Re: How many positive seven-digit integers include the sequence 123, in th [#permalink]
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