How many even 3 digit integers greater than 700 with

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NandishSS: Master | Next Rank: 500 Posts; Posts: 366; Joined: Fri Jun 05, 2015 3:35 amFollowed by: 2 members; Thanked: 3 times

by NandishSS » Fri Sep 16, 2016 5:53 am

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

Source:OG

OA:E

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Source: Beat The GMAT — Problem Solving | Fri Sep 16, 2016 5:53 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Sep 16, 2016 6:19 am

NandishSS 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

Case 1: Hundreds digit is 7 or 9
Number of options for the hundreds digit = 2. (7 or 9)
Number of options for the units digit = 4. (2, 4, 6, or 8)
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply:
2*4*7 = 56.

Case 2: Hundreds digit is 8
Number of options for the hundreds digit = 1. (Must be 8)
Number of options for the units digit = 3. (2, 4, or 6)
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply:
1*3*7 = 21.

Total ways = Case 1 + Case 2 = 56+21 = 77.

The correct answer is E.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Sep 16, 2016 7:34 am

Here's a similar question to practice with: https://www.beatthegmat.com/counting-a-3 ... 88550.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Sat Sep 17, 2016 2:31 pm

GMATGuruNY wrote:
NandishSS 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
Case 1: Hundreds digit is 7 or 9
Number of options for the hundreds digit = 2. (7 or 9)
Number of options for the units digit = 4. (2, 4, 6, or 8)
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply:
2*4*7 = 56.

Case 2: Hundreds digit is 8
Number of options for the hundreds digit = 1. (Must be 8)
Number of options for the units digit = 3. (2, 4, or 6)
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply:
1*3*7 = 21.

Total ways = Case 1 + Case 2 = 56+21 = 77.

The correct answer is E.

Hi Mitch,

I tried to solve it as follows:
Number of options for the hundreds digit = 3. (7 or 8 or 9)
Number of options for the units digit = 3. (2, 4, or 6) - excluding 8
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply: 3*3*7= 63

Where did I go wrong?

Thanks

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Sep 17, 2016 3:16 pm

Mo2men wrote:Hi Mitch,

I tried to solve it as follows:
Number of options for the hundreds digit = 3. (7 or 8 or 9)
Number of options for the units digit = 3. (2, 4, or 6) - excluding 8
Number of options for the tens digit = 7. (Of the 9 remaining nonzero digits, any but the 2 already used)
To combine these options, we multiply: 3*3*7= 63

Where did I go wrong?

Thanks

The red portion miscounts the number of options for the units digit.
The red portion is valid only if the hundreds digit is 8, in which case the units digit can be 2, 4, or 6, yielding a total of 3 options.
But if the hundreds digit is 7 or 9, then the units digit can be 2, 4, 6, or 8, yielding a total of 4 options.
Thus, it is incorrect to say that -- in every case -- the number of options for the units digit = 3.

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