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# How many integers between 2,000 and 6,999 are even and have a digit th

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Joined: 02 Sep 2009
Posts: 59632
How many integers between 2,000 and 6,999 are even and have a digit th  [#permalink]

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03 Jan 2019, 09:31
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Difficulty:

35% (medium)

Question Stats:

68% (02:05) correct 32% (02:01) wrong based on 56 sessions

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How many integers between 2,000 and 6,999 are even and have a digit that is a prime number in the tens place?

(A) 200
(B) 120
(C) 510
(D) 110
(E) 1000

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Posts: 4131
Re: How many integers between 2,000 and 6,999 are even and have a digit th  [#permalink]

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03 Jan 2019, 09:43
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Top Contributor
1
Bunuel wrote:
How many integers between 2,000 and 6,999 are even and have a digit that is a prime number in the tens place?

(A) 200
(B) 120
(C) 510
(D) 110
(E) 1000

Take the task of creating 4-digit integers and break it into stages.

Stage 1: Select a thousands digit
Since the numbers range from 2000 to 6999, the thousands digit can be 2, 3, 4, 5 or 6
So, we can complete stage 1 in 5 ways

Stage 2: Select a hundreds digit
This digit can be any digit from 0 to 9.
We can complete stage 2 in 10 ways

Stage 3: Select a tens digit
This digit must be prime (2, 3, 5, or 7)
We can complete stage 3 in 4 ways

Stage 4: Select a units digit
This digit must be EVEN (2, 4, 6, 8, or 0)
We can complete stage 4 in 5 ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create a 4-digit number) in (5)(10)(4)(5) ways (= 1000 ways)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Re: How many integers between 2,000 and 6,999 are even and have a digit th  [#permalink]

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03 Jan 2019, 09:53
1
Bunuel wrote:
How many integers between 2,000 and 6,999 are even and have a digit that is a prime number in the tens place?

(A) 200
(B) 120
(C) 510
(D) 110
(E) 1000

The thousands digit can be 2/3/4/5/6 (note that all integers with these thousands digit are a part of our range) -> 5 ways
The hundreds place can be any of the 10 digits -> 10 ways
The tens place can be 2/3/5/7 -> 4 ways
The units place must be one of 0/2/4/6/8 -> 5 ways

Total = 5*10*4*5 = 1000 such numbers

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Re: How many integers between 2,000 and 6,999 are even and have a digit th  [#permalink]

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03 Jan 2019, 10:30
from 2000 to 6999
thousand's digiti can have 5 ways: 2,3,4,5,6
hundreds digit: any number from 0 to 9: 10 ways
tens digit: 2,3,5,7: 4 ways
unit digit has to be even so, 0,2,4,6,8: 5 ways
Total ways = 5*10*4*5
1000
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Re: How many integers between 2,000 and 6,999 are even and have a digit th  [#permalink]

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06 Jan 2019, 04:29
1
Bunuel wrote:
How many integers between 2,000 and 6,999 are even and have a digit that is a prime number in the tens place?

(A) 200
(B) 120
(C) 510
(D) 110
(E) 1000

thousandths place : 2,3,4,5,6
hundreds place :10
tens: 2,3,5,7
units: 0,2,4,6,8

5*10*4*5 = 1000
IMO E
Re: How many integers between 2,000 and 6,999 are even and have a digit th   [#permalink] 06 Jan 2019, 04:29
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