Combinations problem - Finding the number I would like to ask for your confirmation to my thought in regards with the following exercise:
One man has 30 different statues, 27 genuine and 3 fake. He sold 10 of these statues in museum A, 10 in museum B and 10 in museum C. What is the probability that each of these museums bought fake statue from the man?
I found the probability for each museum, through combinations, that is :
$$
\frac{{3 \choose 1} {27 \choose 9}+{3 \choose 2 } {27 \choose 8}+{3 \choose 3} {27 \choose 7}}{{30 \choose 10}}
$$
Is my way of thinking correct?
Thank you very much in advance.

Your answer seems wrong. Although, from your answer I am unable to judge about how you exactly arrived at it but here's a hint to how you could have approached the problem:
Just in case you don't know: Number of ways to divide n objects in n1 groups of m1 object, n2 groups of m2 object and so on till nk groups of mk objects such that Σni mi = n can be given as:
$\frac{n!}{(m1 !)^{n1} (n1 !) .... (mk !)^{nk} (nk !)}$
(This result can be arrived at using simple product rules and a bit of intuition or more formally by using set theory and providing appropriate bijections)
You can use this to frame your answer with the following thought process:
Compute total possible ways to distribute( without any restrictions ) and use this as denominator of your probability fraction.
Compute total possible ways to distribute 27 genuine pieces among 3 groups of 9 each. Also, calculate total possible ways to distribute 3 defected pieces among 3 groups of 1 each. And multiply these two computations using product rule to get total number of ways to distribute 30 pieces such that each one gets 9 genuine and 1 defected piece. This gives you the numerator.
Finally, obtain the probability in terms of a fraction.
Hope this helps!
