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Alan and Kevin POTW: How we did this was to use factorials. First we figured out the total number of possibilities/the denominator. Because we know that this can count repeats, this must be a permutation.
First box can have 10 possibilities, and so can the second, third, and fourth. So it is 10*10*10*10 or 10 to the power of 4, which is 10 000.
Now we narrowed down our options for this question. The average between all 4 of the digits is 8.5. This means that the smallest number that can occur is 7, the license plate being 7999. However, you can make different combinations using 7999. For this, we used factorials because it is so much quicker than counting.
Because you can have 4 numbers for the first digit, 3 for the second, and so on, this means that you can use 4! to find the total number of combinations. However, there are repeats of 9s by three times, so we have to divide 4! by 3!, which gives us 4.
To test this, we counted the 7999 possibilities, which are 7999, 9799, 9979, 9997. This would be correct.
Then we did the same thing for 8899, the second possibility for combinations.
4! / 2! / 2!(Since there are 2 repeats of 8 and 9 each.) This would be 24 / 2 / 2 = 6. There is also 8999, which is the same as 7999, so that would be 4 combinations as well.
And finally, there is 9999, and there is only one combination for that.
Adding all of those up, we get 6+4+4+1 = 15. Thus, the answer would be 15/10000 combinations, simplified to 3/2000. Decimal is 0.0015, and percent is 0.15%.
Counting is fine, but fortunately for you Seayrohn this question had only a few different combinations. If there was a way more complicated question than this one with a lot more combinations, then counting would take you forever! XD
Oh yeah btw I accidentally posted another messages using an unknown account... so the other comment in this is mine as well.
By using the largest possible digit 9, I figured a much quicker way of doing this problem. 0-6 with 3 9s will lead to 27-33. Since 9 is the largest possible number, 0-6 can't be used to create 34 or more. From there, there are only 3 digits that can be used. 7, 8, 9. Now, just make combinations that go above 34 or 34. 4 combinations are 7999, 9799, 9979, 9997. That is the only combinations for 7. For 8, there are 10 combinations. The 4 that can be mixed up from before with just 1 8 and 3 9s, and using 2 8s and 2 9s to mix it up. Finally, for the last 9, there is just 9999 giving off a final combo. 4+10+1 is 15. Therefore, there are 15 combinations. There are 10 possible digits for each of the 4 possible digits. All the possible combinations adds up to 10, 000 combos (10x10x10x10). So the answer to this question is that the probability of getting 34 or more when adding all the digits is 15/10000 simplified to 5/2000, 0.0015, and 0.15%.
Sorry for doing it late. I got 3/2000 or 0.0015 or 0.15%. Basically what I did was find all the possibilities with the numbers 7-9 because even if we did 9996 (Or something similar to that with a number that is lower than 7), the sum will be lower than 34. And in total there were 15 possibilities out of 10 000.
Alan and Kevin POTW:
ReplyDeleteHow we did this was to use factorials.
First we figured out the total number of possibilities/the denominator.
Because we know that this can count repeats, this must be a permutation.
First box can have 10 possibilities, and so can the second, third, and fourth.
So it is 10*10*10*10 or 10 to the power of 4, which is 10 000.
Now we narrowed down our options for this question.
The average between all 4 of the digits is 8.5. This means that the smallest number that can occur is 7, the license plate being 7999.
However, you can make different combinations using 7999.
For this, we used factorials because it is so much quicker than counting.
Because you can have 4 numbers for the first digit, 3 for the second, and so on, this means that you can use 4! to find the total number of combinations. However, there are repeats of 9s by three times, so we have to divide 4! by 3!, which gives us 4.
To test this, we counted the 7999 possibilities, which are 7999, 9799, 9979, 9997. This would be correct.
Then we did the same thing for 8899, the second possibility for combinations.
4! / 2! / 2!(Since there are 2 repeats of 8 and 9 each.)
This would be 24 / 2 / 2 = 6.
There is also 8999, which is the same as 7999, so that would be 4 combinations as well.
And finally, there is 9999, and there is only one combination for that.
Adding all of those up, we get 6+4+4+1 = 15.
Thus, the answer would be 15/10000 combinations, simplified to 3/2000.
Decimal is 0.0015, and percent is 0.15%.
I just counted...
DeleteCounting is fine, but fortunately for you Seayrohn this question had only a few different combinations. If there was a way more complicated question than this one with a lot more combinations, then counting would take you forever! XD
DeleteOh yeah btw I accidentally posted another messages using an unknown account... so the other comment in this is mine as well.
I would've found an easier way.
Deletepossibly
also, just saying. I'll still get the answer from counting eventually...
Deleteeither way, this'll be very helpful for the future (I'm talking about your method Alan instead of counting)
DeleteBy using the largest possible digit 9, I figured a much quicker way of doing this problem. 0-6 with 3 9s will lead to 27-33. Since 9 is the largest possible number, 0-6 can't be used to create 34 or more. From there, there are only 3 digits that can be used. 7, 8, 9. Now, just make combinations that go above 34 or 34. 4 combinations are 7999, 9799, 9979, 9997. That is the only combinations for 7. For 8, there are 10 combinations. The 4 that can be mixed up from before with just 1 8 and 3 9s, and using 2 8s and 2 9s to mix it up. Finally, for the last 9, there is just 9999 giving off a final combo. 4+10+1 is 15. Therefore, there are 15 combinations. There are 10 possible digits for each of the 4 possible digits. All the possible combinations adds up to 10, 000 combos (10x10x10x10). So the answer to this question is that the probability of getting 34 or more when adding all the digits is 15/10000 simplified to 5/2000, 0.0015, and 0.15%.
ReplyDeleteI somehow thought 15 divided by 5 is 5 and didn't notice. It's actually 3/2000.
DeleteIs there going to be POTW for the last week of school or not because school is over
ReplyDeleteI'll put one up as long as some people are interested in doing it. maybe I'll select a Grade 12 question just for fun or curiosity!
Delete11/9999/ 1099.89%/ 0.1099.89 .Work done in notebook.
ReplyDelete13/6561. 9*9*9*9=6561, and there are 13 combinations that are 34 or above.
ReplyDeleteThe probability of getting a combination 34 or above is 15/10000, 0/15% or 0.0015. I did the work on scrap paper
ReplyDeleteSorry for doing it late.
ReplyDeleteI got 3/2000 or 0.0015 or 0.15%.
Basically what I did was find all the possibilities with the numbers 7-9 because even if we did 9996 (Or something similar to that with a number that is lower than 7), the sum will be lower than 34. And in total there were 15 possibilities out of 10 000.