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

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Math Expert
Joined: 02 Sep 2009
Posts: 60605
How many positive seven-digit integers include the sequence 123, in th  [#permalink]

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26 May 2018, 07:55
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Difficulty:

95% (hard)

Question Stats:

19% (02:09) correct 81% (02:27) wrong based on 102 sessions

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

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26 May 2018, 08:22
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Bunuel wrote:
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

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.
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Manager
Joined: 25 Jul 2017
Posts: 92
How many positive seven-digit integers include the sequence 123, in th  [#permalink]

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26 May 2018, 08:34
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1
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.

Manager
Joined: 10 May 2018
Posts: 119
Concentration: Finance, Sustainability
Re: How many positive seven-digit integers include the sequence 123, in th  [#permalink]

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22 Jul 2018, 22:10
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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Joined: 25 Jul 2017
Posts: 92
Re: How many positive seven-digit integers include the sequence 123, in th  [#permalink]

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15 Nov 2019, 05:14
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.

Re: How many positive seven-digit integers include the sequence 123, in th   [#permalink] 15 Nov 2019, 05:14
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