The number of ways of getting three of a kind $(x, x, x, y, z)$ when rolling 5 dice is -
${5 \choose 3}{2 \choose 1}{6 \choose 1}{5 \choose 2}$
${5 \choose 3}$ - Ways of choosing dice for the $x$'s
${2 \choose 1}$ - Ways of choosing dice for the $y$
${6 \choose 1}$ - Ways of choosing the number for the $x$'s
${5 \choose 2}$ - Ways of choosing the numbers for the $y$ and $z$
I am looking for the intuition behind why we can't have $6 \choose 3$ for the ways of choosing the numbers for $x, y, z$. I actually know why it is but I could really use a clear intuitive explanation so my head isn't wrecked with it. I want to be able to visualize it quickly and clearly.