Determine in how many ways you can select 5 cards from a standard 52-card bundle, and each color is chosen among your selections.
The standard set of cards contains 13 different cards in 4 colors.
I am looking for solution of this.
My idea is
$${\binom{13}{1} \binom{13}{1} \binom{13}{1} \binom{13}{2} }$$
Disclaimer: this is not a homework
I think you are almost there. One more thing you should consider is that the color including $2$ cards can be chosen from $4$ different colors as well (which color will have $2$ cards?). This makes the answer $$\binom{4}{1}\binom{13}{1} \binom{13}{1} \binom{13}{1} \binom{13}{2}$$