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How many even three-digit integers have the property that their digits

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Math Expert
Joined: 02 Sep 2009
Posts: 54376
How many even three-digit integers have the property that their digits  [#permalink]

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26 Mar 2019, 05:38
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Difficulty:

75% (hard)

Question Stats:

33% (02:52) correct 67% (02:36) wrong based on 21 sessions

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How many positive even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

(A) 21
(B) 34
(C) 54
(D) 72
(E) 150

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Re: How many even three-digit integers have the property that their digits  [#permalink]

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26 Mar 2019, 05:58
1
Bunuel wrote:
How many positive even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

(A) 21
(B) 34
(C) 54
(D) 72
(E) 150

I am assuming "strictly increasing order" means the digits cannot be equal.

For the numbers to be even, the units digit must be 0/2/4/6/8. But it must be the greatest digit too. So it cannot be 0 or 2.

Units digit 4:
The two leftmost digits can be 1/2/3. Two can be chosen in 3C2 ways and arranged in only 1 way (smaller on left and greater in right)
So 3 ways

Units digit 6:
The two leftmost digits can be 1/2/3/4/5. Two can be chosen in 5C2 ways and arranged in only 1 way (smaller on left and greater in right)
So 5*4/2 = 10 ways

Units digit 8:
The two leftmost digits can be 1/2/3/4/5/6/7. Two can be chosen in 7C2 ways and arranged in only 1 way (smaller on left and greater in right)
So 7*6/2 = 21 ways

Total = 3 + 10 + 21 = 34 ways

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Re: How many even three-digit integers have the property that their digits  [#permalink]

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26 Mar 2019, 06:13
Assume that the first digit is 1 and second digit is 2 then we have 3 possible third digit. While first digit remains 1 second changes to 3 then we have 3 possible third digit. So with first digit 1 we have 3,3, 2,2,1,1,possible numbers when we change first digit to 2 we will have 3,2,2,1,1,possible numbers and so on. So 12+9+6+4+2+1=34 option B.

Is there a shorter way to work it out?

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Re: How many even three-digit integers have the property that their digits  [#permalink]

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26 Mar 2019, 11:48
1
Bunuel wrote:
How many positive even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

(A) 21
(B) 34
(C) 54
(D) 72
(E) 150

even digits ending with 0,2,4,6,8
so 0,2 wont be possible as all three digits have to be unique
so
case 1
ending with 4
first two digits can be 1,2,3 ; 3c2 3
ending with 6 ; 1,2,3,4,5; 5c2 ; 10
ending 8 ; 1,2,3...7 ; 7c2 ; 21
total 34
IMO b
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Re: How many even three-digit integers have the property that their digits   [#permalink] 26 Mar 2019, 11:48
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