Answer:
The total number of different arrangements is 560.
Step-by-step explanation:
A multiset is a collection of objects, just like a set, but can contain an object more than once.
The multiplicity of a particular type of object is the number of times objects of that type appear in a multiset.
Permutations of Multisets Theorem.
The number of ordered n-tuples (or permutations with repetition) on a collection or multiset of
objects, where there are
kinds of objects and object kind 1 occurs with multiplicity
, object kind 2 occurs with multiplicity
, ... , and object kind
occurs with multiplicity
is:
![\begin{equation*}\frac{n!}{n_1!*n_2!*\dots * n_k!}\end{equation*}](https://tex.z-dn.net/?f=%5Cbegin%7Bequation%2A%7D%5Cfrac%7Bn%21%7D%7Bn_1%21%2An_2%21%2A%5Cdots%20%2A%20n_k%21%7D%5Cend%7Bequation%2A%7D)
We know that a boy has 3 red, 2 yellow and 3 green marbles. In this case we have n = 8.
If marbles of the same color are indistinguishable, then the total number of different arrangements is
![{8 \choose 3, 2, 3} = \frac{8 !}{3 ! 2 ! 3 !} = \frac{8\cdot \:7\cdot \:6\cdot \:5\cdot \:4}{2!\cdot \:3!}=\frac{6720}{2!\cdot \:3!}=\frac{6720}{12}=560](https://tex.z-dn.net/?f=%7B8%20%5Cchoose%203%2C%202%2C%203%7D%20%20%3D%20%5Cfrac%7B8%20%21%7D%7B3%20%21%202%20%21%203%20%21%7D%20%3D%20%5Cfrac%7B8%5Ccdot%20%5C%3A7%5Ccdot%20%5C%3A6%5Ccdot%20%5C%3A5%5Ccdot%20%5C%3A4%7D%7B2%21%5Ccdot%20%5C%3A3%21%7D%3D%5Cfrac%7B6720%7D%7B2%21%5Ccdot%20%5C%3A3%21%7D%3D%5Cfrac%7B6720%7D%7B12%7D%3D560)