Three numbers are selected without replacement from {1,2,3,4,5,6,7} to form a 3-digit number. What is the probability that the number is greater than 537?

Answer 1

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The first digit Must be either 5,6,7

IF The first digit is 6 or 7 then the outcome

is of interest regardless of the other two digits..

The probability of that happening is 2/7

If the first digit is 5, then the second digit MUST

be 4,6, or 7.

The probability of that happening is 1/7* 3/6 = 1/14

the final probability is 2/7 + 1/14 = 5/14

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combinatorically:

The denominator is (7 pick 3) =

7! / (7-3)! =

7! / 4!

= 7*6*5

= 210

if the first digit is 6, then there are

6 ways to choose the 2nd digit, and 5 to choose the 3rd,

for a total of 30.

the same argument is made if the first digit is 7.

So there are 60 ways to choose in this case.

But if the first digit is 5, then there are only 3 ways

to choose the 2nd digit, and then 5 ways to choose the 3rd.

So in this case, there are 15 ways to choose...

(60+15)/210 = 75/210 = 15/42 = 5/14