hard probability question from a book

In a factory, we know that 20% of produced goods will be replaced. How many products will have to be produced to the warehouse, so that 10 000 orders and the replacements can be sent be with probability of 0.9 ? The answer is 12 051, but I dont know how to calculate it, or even understand the question properly

$$\small{\text{binomcdf$\ (10000,0.2,\textcolor[rgb]{1,0,0}{2051}) \\\\ = \sum \limits_{i=0}^{\textcolor[rgb]{1,0,0}{2051}} \binom{10000}{ i}*0.2^i*0.8^{(10000-i)} =0.9008 \\ $ }}$$

You can calculate it with www.wolframalpha.com and input "sum ( binom(10000,i)*0.2^i*0.8^(10000-i) ) from i=0 to 2051"

10000 + 2051 = 12051       12051 products will have to be produced to the warehouse.

This is what I came up with but even Wolfram|alpha said it was too difficult to calculate

$$0.9=\frac{X!}{10000!(X-10000)!}*0.2^{10000}*0.8^{X-10000}$$

To be honest I am not even sure that i understand the question properly either.   

I have put a reminder post on this one because I am interested too.   

http://web2.0calc.com/questions/unanswered-question-probability-i-am-interested-too

Thanks Heureka,

I have interpreted this question the other way around I think.  

I think the wording is very poor.

Even so I need to think about it.

Since your answer is the same as the 'book'.  It is obviously the interpretation that they wanted. :/

I interpreted this differently, so I don't get the book's answer. I said the production line turns out products of which 20% are defective. We want to test and discard sufficient numbers of defective items from a batch so as to leave the remainder with just 10% defective items. We need to produce sufficient to cover this procedure as well as having 10,000 plus the expected 1000 replacements that will be demanded. (90% quality control, so 10% will be defective, and returned.)

I have not yet allowed for replacements of the replacements.

Anon, you interpreted this the same way as I did.

I think that formula I wrote may be correct but I have not idea how to solve it.

Even Wolfram|Alpha did not solve it !

I think the question was very poorly written.