the answer seems correct.

it should be 12! x 11! x 10!, to represent the possibilities that the 3 required fruits can be picked from. then multiply it by 9C3 to represent the remaining fruits.

The exclaimation point ! in the formula represents the term "Factorial"

From the dictionary

The product of all the positive integers from 1 to a given number: 4 factorial, usually written 4!, is equal to 24 (1 × 2 × 3 × 4 = 24).

But,9P3 would mean that each of the 3 boys would get same no. of fruits,1 of the 9 fruits precisely.(maybe )However the question permits a boy to get a max. no. of 10 fruits. so,distributions like 10-1-1,    9-2-1 etc  don't come in consideration.  


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To be clearer,9P3 would mean the no. of ways you can give 1 fruit (out of of 9 diiff. fruits) to each of the 3 boys.  

First of all,if a boy gets the remaining 9 fruits,then 3 outcomes are possible(any of the 3 boys can get it).And it will get more difficult to calculate the other possible distributions.

Secondly,how can you be so sure that the no. of distributions will follow a fixed rule like 3 for 10-1-1, so x for 9-2-1 and so on.

Thirdly,I don't get your last line.(can't use quotes )

LASTLY,, I know the correct procedure to solve the problem as I have said earlier. What I want is to know the error in my procedure(1st post)


 

Well then find the error in the book's solution.Here it is-

"No. of distributions possible(without any restrictions) is
3^12.----------(1)

No. of cases where a boy can get all the 12 fruits=3C1(any of the 3 boys can get them).Alternately,we can say that in 3 cases 2 boys won't get any fruit.-------(2)

now,if a boy gets no fruit then the other two will get them.No. of distributions possible for 2 boys=2^12.Here also a boy (of the 2) can get no fruit in 2 ways.So, no. of distributions where the 2 boys would get 12 fruits with each having at least one fruit = 2^12-2.
          so,1 out of the 3 boys would get no fruit in 3C1(2^12 -2) ways--------------------(3)

From (1),(2) & (3) we get,Required no. of distributions
=Total distrbutions-(2 boys get no fruit + 1 boy gets    
                                                       no fruit)
=3^12-{3+3C1(2^12-2)}
=519156.
           Ans.

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I am starting to think there may be 2 correct answers.  

i still prefer your solution  

thanx for that.But do you mean there ARE two correct answers or are you tired of "finding the error"?

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Hey Mitch ,i think i should expect some suggestion from you.You are the moderator of this forum.