I feel confused about selecting suits and cards that each suit corresponds to.

Suppose we have a standard deck of cards, and we want to form a 6 hands with at most 3 suits.

I understand this needs to be spliced into 3 cases to discuss, which are 1 suit case, 2 suits case, and 3 suits case.

But I feel confused when it comes to select cards.

For example, I am not sure if I understand this correctly, but I feel like if we only need to form cards with a single suit, then this can be done by $4(13 * 12 * 11 * 10 * 9 * 8)$ ways, since each suit has 13 cards, and we have 4 suits here.

And when it comes to the 2 suits case, I feel like we only need to draw from the total of 26 cards to form such hands, and there are 6 ways to do this, but I am not sure if my understanding about the 26 cards is right or what.

And for the 3 suits case, I feel like it's just choose 3 suits out of 4, and draw cards from total of 39 cards.

Is my understanding towards the total number of cards in each case correct? If not, why so?