Brain Teasers
4 X 4
Below there are sixteen numbers. Assuming that any three of the numbers may be drawn at random, what are the chances that three numbers will be drawn whose sum equals six?
1 2 3 1
2 3 1 2
3 1 2 3
1 2 3 1
1 2 3 1
2 3 1 2
3 1 2 3
1 2 3 1
Answer
This is a bit tricky and I initially thought the answer was 1/16 or 6.2% but it is really 2/7 or 28.6%.Either drawing '2 2 2' or some combination of '1 2 3' adds up to be 6. The chances of getting '2 2 2' are 5/16 (5 2s out of 16) times 4/15 (4 2s left out of 15) times 3/14 (3 2s left out of 14). This is 5/16 * 4/15 * 3/14 which equals 1/56.
The chances of getting some combination of '1 2 3' is equal to 6/16 (6 1s in the list) times 5/15 (5 2s out of 15 numbers left) times 5/14 (5 3s out of 14 numbers left) times the number of different combinations of '1 2 3' which is 6. This is 6/16 * 5/15 * 5/14 * 6 which equals 15/56.
1/56 + 15/56 = 16/56 or 2/7 or ~28.6%.
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Comments
Very good!
Another approach to the same math -
Ways to put together 1 2 3 are 6 * 5 * 5 = 150. Because they are distinguishable, we have no permutations of order.
Ways to put together 2 2 2 are 5! / (3! * 2!) = (5 * 4) / 2 = 10.
Ways to pick combinations of three items from a group of 16 are 16! / (3! * 13!) = (16 * 15 * 14) / (3 * 2) = 16 * 5 * 7 = 560.
Another approach to the same math -
Ways to put together 1 2 3 are 6 * 5 * 5 = 150. Because they are distinguishable, we have no permutations of order.
Ways to put together 2 2 2 are 5! / (3! * 2!) = (5 * 4) / 2 = 10.
Ways to pick combinations of three items from a group of 16 are 16! / (3! * 13!) = (16 * 15 * 14) / (3 * 2) = 16 * 5 * 7 = 560.
I would also matter How many numbers one can pick out at one time..
Disregard that comment I didn't read entirely >.> Very good
Gah, I should've gotten it. Multiplied by 3 instead of 6 because I didn't think it all the way through, so I ended up with a little over 15% instead of 28%. Nice one, thought it would be a joke at first, then realized it required a bit of thought.
This belongs in probability.
I was going to say that, shadows, before I saw your comment. Overall good teaser
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