Packaging of virus particles (Probability)

I am currently reading a virology textbook and am stumped by, what I assume, is a problem that can be solved using probability calculations. It is as follows:

Influenza virus has genetic material which is split into 8 parts which are called vRNPs. Each vRNP is distinct, and each part must be replicated and packaged into a new virus particle for progeny virus to be infectious. In other words, if a newly made virus does not contain vRNP1, vRNP2 . . vRNP8 it wont function correctly. More than eight vRNPs can be packaged though and so duplication may occur (e.g. you may have several of vRNP 1) and importantly packaging is random so their may any assortment of vRNPs.

If influenza virus particles were to randomly package 12 vRNPs, then 10% of those particles would have a complete set of genomic segments and be infectious.


I do not understand how to calculate the above. How has this conclusion been arrived at?

Any help is very much appreciated.

J

Most likely someone employed an approximation to the multinomial distribution. it is possible to calculate such a probablility exactly using a computer program, but it is somewhat involved. It is essentially a problem in combinatorics.

I inferred from other calculations in the book that this problem would be fairly straight forward and can be worked out using a calculator; in fact, the author specifically encourages you to spend a few minutes working this particular one out yourself. Is a computer program really necessary?

You need to evaluate the following: \(\displaystyle \sum 12!/x_1!...x_8!\) where the sum goes over all possible combinations so that each \(\displaystyle x_i\) is at least 1, and \(\displaystyle \sum x_i\) is at least 8. Then multiply the result by \(\displaystyle (1/8)^{12}\). For example, there are 8 ways that exactly one \(\displaystyle x_i\) contains 2, with the other 7 containing 1, for a total of 9. Yes, it can be done on a calculator if you are careful.