GRE Quant Combinatorics problem The University of Maryland, University of Vermont, and Emory University have each $4$ soccer players. If a team of $9$ is to be formed with an equal number of players from each university, how many number of ways can the selection be done?

Possible answers are: $3, 4, 12, 16, 25$.

I have encountered that problem in GRE Quant. 

Here is my solution: From each university we should take $9:3=3$ players and this choice could be done with $C_4^3=4$ ways. Hence the selection can be done in $4\times 4 \times 4=4^3=64$ ways. But such answer does not exist.

Can anyone explain this moment please?
I'd frame exactly the same computation you use as "the number of ways to pick the one player that is excluded from each University ...".  Of course, I'd get the same answer...

As a quick enumeration of at least 26 different ways to make some of these teams...  

Let $a, b, c, d$ represent the four players from the University of Maryland.  Let $f, g, h, i$ represent the four players from the University of Vermont.  Let $p, q, r, s$ represent the four players from Emory University.  The question is this ungrammatical mess "If a team of 9 is to be formed with an equal number of players from each university, how many number of ways can the selection be done?"  In particular, "how many number of ways" is not grammatically correct English.  Nevertheless,  we will represent a "selection" by who from each University is not selected (since listing the three selected players is equivalent to listing the one non-selected player from each University).  \begin{align*}
afp &amp;&amp; afq &amp;&amp; afr &amp;&amp; afs  \\
agp &amp;&amp; agq &amp;&amp; agr &amp;&amp; ags  \\
ahp &amp;&amp; ahq &amp;&amp; ahr &amp;&amp; ahs  \\
aip &amp;&amp; aiq &amp;&amp; air &amp;&amp; ais  \\  \\
bfp &amp;&amp; bfq &amp;&amp; bfr &amp;&amp; bfs  \\
bgp &amp;&amp; bgq &amp;&amp; bgr &amp;&amp; bgs  \\
bhp &amp;&amp; bhq &amp;&amp; &amp;&amp;\text{+2 more}  \\
&amp;&amp;&amp;&amp;\text{+4 more} \\  \\ 
&amp;&amp;&amp;&amp;\text{+16 more} \\  \\ 
&amp;&amp;&amp;&amp;\text{+16 more} \\  \\ 
\end{align*}
I cross check with other answer is 12=4+4+4

The question is asking selection for each university hence Principal of Addition will apply. Principal of Multiplication will be applied when asked about number of possible combination of team.
The question specifically asks selection from each university not selection of team