Finding the sample space of an experiment Suppose that a study is being done on all families with one, two, or three children. Let the
outcomes of the study be the genders of the children in descending order of their age. List the sample space. 
This my solution: {M, F, MF, FM, MM, FF, MFM, MMF, FMM, MFF,FFM, FMF, FFF, MMM}
I am not sure because there could be more events if two 'M's or 'F's are different. For example, $M_1M_2F$ and $M_2M_1F$ are two different events.
 A: The random experiment is to simply observe a family with $1$ to $3$ children and jot down the "gender of the siblings in the order of their age." You could have decided to write down the sum of the ages of the children, or the color of their eyes. It is random because you can't predict what the individual outcome will be for a particular family, until, of course, you meet the children.
But what you know is the sample space, $\Omega$, such that each individual outcome $\omega \in \Omega$.
You can assign a random variable as function from these outcomes to the real line. Or not.
This is all very consistent with the introductory part of your question, i.e. the first paragraph.
In the second paragraph you introduce the concept of event. This concept is unnecessary in the definition of the sample space. In the case of a finite sample space the sigma-algebra is typically $2^\Omega$, so slipping into events is natural, but unnecessary, and probably prone to mistakes.
In addition, you seem to be confusing the aseptic definition of the random experiment (you take a look at the kids, note their gender, place the genders in order of age) with other details that are not included in your definition of the experiment. If the Smiths have adorable children, Matt and Mary, and Matt is older, the outcome is just as much $\{\bf MF\}$ as in the case of the Miller's, who have horrible children, Paul and Lucy, much older than Mary and Matt, but also with the Paul older than Lucy. 
I think that in your question what you mean is that each child is different from the other (?), but it doesn't matter, as long as the respective genders and age relation stays the same, the outcome is indistinguishable.
A: it says "in descending order of their age". so you dont need to use numbers with the letters. it would look like this. 
{M, F, MF, MM, FM, FF, MMM, MMF, MFM, MFF, FFF, FFM, FMF, FMM}
