In American Chinese restaurants, the standard "dessert" is a "fortune cookie" -- a thin, crisp shell inside which is placed a small slip of paper, usually containing an inspirational message of some sort (rather than any kind of prophecy). Sometimes on the reverse of the slip of paper there is a Chinese word or phrase, along with six "lucky numbers" from 1 to 60 (inclusive).

If you were to open three fortune cookies, the expected number of
*
distinct
*
lucky numbers you'd get can be written as
$\frac{a}{b}$
, where
$a$
and
$b$
are positive coprime integers. Find
$a+b$
.

**
Details and assumptions
**

- Just as an example, the three fortunes pictured have 13 distinct lucky numbers.
- The lucky numbers are distributed uniformly.
- The six numbers on each fortune are distinct.

The answer is 863.

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The probability that 1 does not appear on any fortune is $(9/10)^3 = 729/1000$ , so the probability that 1 is among the lucky numbers is $271/1000$ . The same holds for each of the 60 possible lucky numbers, so by linearity of expectation the number of values from 1 to 60 which appear in some fortune is $60*271/1000 = 813/50$ . The answer is $813+50=863$ .