Probability

Question

The nine digits 1,2,3,4,5,6,7,8,9 are arranged in random order to form a 9 digit number. Find the probability that 1,2 and 3 appears as neighbours in the order mentioned.[spoiler] answer 1/72 [/spoiler]

GMATGuruNY · Answer

Karishmad2209 wrote:The nine digits 1,2,3,4,5,6,7,8,9 are arranged in random order to form a 9 digit number. Find the probability that 1,2 and 3 appears as neighbours in the order mentioned.[spoiler] answer 1/72 [/spoiler]

There are a total of 9 positions in the integer.
For 1, 2 and 3 to appear in CONSECUTIVE ORDER, the "2" must occupy one of the 7 MIDDLE POSITIONS in the integer.

P("2" occupies one of the 7 middle positions) = 7/9. (Of the 9 positions in the integer, 7 are in the middle.)
P("1" is immediately to the LEFT of 2) = 1/8. (Since there are 8 remaining digits that could be placed immediately to the left of 2)
P("3" is immediately to the RIGHT of 2) = 1/7. (Since there are 7 remaining digits that could be placed immediately to the right of 2.)
To combine these probabilities, we multiply:
7/9 * 1/8 * 1/7 = [spoiler]1/72[/spoiler].

Rich.C@EMPOWERgmat.com · Answer

Hi Karishmad2209,

This is a badly-worded question. Mitch's explanation is correct, IF the digits 1, 2 and 3 must appear in THAT SPECIFIC ORDER somewhere in the 9 digit number. However, the question isn't worded with that restriction. Stating that the numbers 1, 2 and 3 "must appear as neighbors" COULD mean that the 3 numbers could appear in any of the following orders: 123, 132, 213, 231, 312, 321 - this would change the number of possibilities.

Official GMAT questions are always meticulously worded, so that there can be no confusion or misinterpretation of what the question is asking for. You should be suspicious of wherever this question came from. You might be working with materials that are not realistic.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

GMATGuruNY · Answer

Rich.C@EMPOWERgmat.com wrote:Hi Karishmad2209, This is a badly-worded question. Mitch's explanation is correct, IF the digits 1, 2 and 3 must appear in THAT SPECIFIC ORDER somewhere in the 9 digit number. However, the question isn't worded with that restriction. Stating that the numbers 1, 2 and 3 "must appear as neighbors" COULD mean that the 3 numbers could appear in any of the following orders: 123, 132, 213, 231, 312, 321 - this would change the number of possibilities. Official GMAT questions are always meticulously worded, so that there can be no confusion or misinterpretation of what the question is asking for. You should be suspicious of wherever this question came from. You might be working with materials that are not realistic. GMAT assassins aren't born, they're made, Rich I agree that the problem could be worded much better. That said, as neighbors in the order mentioned seems to clearly imply that 1, 2 and 3 must be in consecutive order.

Quasar Chunawala · Answer

Probability = Favourable outcomes/Total outcomes Think of (1,2,3) as one whole. That way, you have 7 items. No. Of arrangements = 7! Total no. Of arrangements = 9! Probability = 7!/9! = 1/72.

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