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How many even 3 digit integers greater than 700 with distinct non zero

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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 04 Jul 2015, 10:14
twixt wrote:
How many even 3 digit integers greater than 700 with distinct non zero digits are there ?

A. 729
B. 243
C. 108
D. 88
E. 77


There are only Three possible cases

In each case Hundreds places is fixed so we fill the unit's place first in order to make the number even and at last we fill the Ten's digit place

Case 1: 7 __ __ When Hundreds digit is 7
7 _7 choices (remaining 7 out of 9 non-zero digits)_ * _4 choices (2,4,6,8)_ = 28 Numbers[/b]

Case 2: 8 __ __ When Hundreds digit is 8
8 _7 choices (remaining 7 out of 9 non-zero digits)_ * _3 choices (2,4,6)_ = 21 Numbers[/b]

Case 3: 9 __ __ When Hundreds digit is 9
9 _7 choices (remaining 7 out of 9 non-zero digits)_ * _4 choices (2,4,6,8)_ = 28 Numbers[/b]

Total Such Numbers = 28 + 21 + 28 = 77 Numbers

Answer: Option E
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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 11 Jul 2016, 06:41
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twixt wrote:
How many even 3 digit integers greater than 700 with distinct non zero digits are there ?

A. 729
B. 243
C. 108
D. 88
E. 77

Let us try for an approximate answer since the values in the choices are far apart by relaxing some constraints.

The last digit can change in 4 ways
The middle digit can change in 8 ways
The first digit can change in 3 ways

This is 4*8*3 = 96 only even with relaxation of constraints. Choices A,B and C can be eliminated

There are some more numbers which will not be included.

when first digit is 7, second digit cannot be 7. So the possibilities reduce to 4*8*2 + 4*7*1=92
Similarly when first digit is 9, second digit cannot be 9 and the possibilities reduce by 4 more = 88.

The number of possibilities will be less than 88 because when first digit is 8, there are other impossible numbers.

So now we can pick the answer as E
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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 12 Apr 2017, 18:10
Quickly decide to use brute force. Count the number of matching cases in each group of ten numbers.

70 => 0
71 => 4
72 => 3
73 => 4
74 => 3
75 => 4
76 => 3
77 => 0
78 => 3
79 => 4
80 => 0
81 => 3
82 => 2
83 => 3
84 => 2
85 => 3
86 => 2
87 => 3
88 => 0
89 => 3
90 => 0
91 => 4
92 => 3
93 => 4
94 => 3
95 => 4
96 => 3
97 => 4
98 => 3
99 => 0

4*8 + 3*13 + 2*3 = 77

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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 20 May 2017, 21:37
Top Contributor
twixt wrote:
How many even 3 digit integers greater than 700 with distinct non zero digits are there ?

A. 729
B. 243
C. 108
D. 88
E. 77

These types can be handled by considering the number of possibilities for each digit. Start with the one with the most constraints.
1. Last digit can be 2,4,6,8 when the first digit is 7 or 9 and 2,4,6 when the first digit is 8. So there are 4 possibilities in the first case and 3 possibilities in the second case for the last digit.
2. Middle digit cannot be one of the first and last digits and cannot be 0. So there are 7 possibilities
3. When first digit is 7 or 9, then we have (4*7) + (4*7)=56
4. When first digit is 8, we have 3*7=21
5. Total is (3)+(4)= 77
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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 09 Jun 2017, 23:35
[E]



Lets go digit wise:

A B C, A is the hundredth digit and C is the ones digit.

As ABC is a three digit greater than 700, A can have values 7,8,9 ( 700<ABC<999)
As ABC is a even number then, C can take values 2,4,6,8

Now Imagine A is 7, and C can take 2,4,6,8 for each of these possibilities of C there are 7 different values B can take up (except 7,0,and the one in C as the digits are unique)

Therefore for A = 7, there are 4*7 = 28 numbers
for A = 8 there are 3*7 (as 8 is already used and C can take only 2,4,6 as its even) = 21
for A = 9 there are 4*7 = 28 numbers

Adding, 28+21+28 = 77
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Re: How many even 3 digit integers greater than 700 with distinct non zero [#permalink]

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New post 10 Jun 2018, 01:15
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Re: How many even 3 digit integers greater than 700 with distinct non zero   [#permalink] 10 Jun 2018, 01:15

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