permutation with repetition? (5-digit ID: prob of >= 3 6s)

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roadrunner

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I am struggling with a problem that I feel I have at least partially solved but I think there is an easier way.

A teacher wants to assign students 5 digit id's. What is the probability that a student will get an id with at least three 6's?

I think five 6's is 1/100,0000

I think four 6"s is 45/100,000

and I am relatively confident in those answers and how I got them.

But for three 6's I think the answer is 810/100,00 and can eventually explain it (at first I thought it might be 81/100,00), reexplain it to myself the way I did four 6's, but I think there is an easier way to get the number. I have tried using permutations but get much larger numbers. So my answers don't even agree.

I hope I have some of it right.

Suggestions?
 
Re: permutation with repetition?

For exactly three sixes, there \(\displaystyle 5 \choose 3\) places for the 6's and \(\displaystyle 9^2\) choices for the other two places.
So 810 is correct.

There are too many zeros in your first answer. It’s just a typo.
Otherwise your work is correct.
 
Re: permutation with repetition?

roadrunner said:
I am struggling with a problem that I feel I have at least partially solved but I think there is an easier way.

A teacher wants to assign students 5 digit id's. What is the probability that a student will get an id with at least three 6's?

I think five 6's is 1/100,0000<<< Incorrect - please show steps to explain how you did get it!

I think four 6"s is 45/100,000 <<<Are you considering exactly four or at least four - please show steps to explain how you did get it!

and I am relatively confident in those answers and how I got them.

But for three 6's I think the answer is 810/100,00 and can eventually explain it (at first I thought it might be 81/100,00), reexplain it to myself the way I did four 6's, but I think there is an easier way to get the number. I have tried using permutations but get much larger numbers. So my answers don't even agree.

I hope I have some of it right.

Suggestions?
Click to expand...
 
I know that the denominator in #1 is 100,000

I am considering exactly 4.

Then adding the three answers for the final answer.
 
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