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Three from Nine (Posted on 2006-04-25) Difficulty: 3 of 5
Nine marbles numbered 1 to 9 are placed in a barrel and three are drawn out, without replacement. Determine -:

1. The probability that the three digit number formed from the marbles in the order drawn is divisible by (a) Five (b) Seven (c) Nine.
2. The probability that a three digit number can be formed by rearranging the marbles drawn, that is divisible by (a) Five (b) Seven (c) Nine.

No Solution Yet Submitted by Vernon Lewis    
Rating: 3.5000 (4 votes)

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Solution Solution | Comment 11 of 12 |

The first premis is that there are 504 possible 3-digit results (9!/6!).  The second is that for any unique 3-digit combination there are 6 permutations (3!) .   I hope my stats knowledge serves me here because these values are used as the basis for some of the following answers.

1a) Only numbers ending in the digit 5 will be divisible by 5.  Therefore the answer is equal to the probability that the last marble drawn will be the digit 5.  (8/9 x 7/8 x 1/7 = 11.1%)

2a) If you are allowed to rearrange, then it is the probability that the digit 5 will be drawn first, second or third.  This will be the inverse of the probability that it will not be drawn at all. (1 - 8/9 * 7/8 * 6/7 = 33.3%)

1c) and 2c) For divisibility by 9, being able to rearrange the result makes no difference, since the only criterion is that the sum of the digits drawn be a multiple of 9.  There are 10 such 3-digit combinations (shown below) which allows for 60 permutations.  60 / 504 = 11.9%
1+2+6 = 9
1+3+5 = 9
2+3+4 = 9
1+8+9 = 18
2+7+9 = 18
3+6+9 = 18
3+7+8 = 18
4+5+9 = 18
4+6+8 = 18
5+6+7 = 18

1a) Sorry, the only approach I had to this one was brute force.  There are 72 3-digit numbers that are divisible by 7 and do not contain the digit 0 or any duplicate digits.  72 / 504 = 14.3%

1b) Of the 72 possible numbers from 1a, there are only 54 unique digit combinations.  The ability to rearrange the results allows for 324 permutations.  324 / 504 = 64.3%.  That can't be right, can it?!  Hmph, I guess it could be.

Edited on April 25, 2006, 6:04 pm

Edited on April 25, 2006, 6:20 pm
  Posted by Kevin on 2006-04-25 17:56:00

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