We have a condensed deck of 40 cards containing only the denominations from Ace through 10.
Presumably that is those ten denominations in each of the four standard suits.
We draw four cards uniformly at random (without replacement).
What is the probability that we draw four distinct denominations, such as one Ace, one five, one six and one seven.
Well, we are comparing the count for ways to select four from ten denominations, with each in one from four suits, to the count for ways to select four from forty cards. That "each" means we are not multiplying ${^{4}\mathrm C_1}$ by $4$ but rather empowering it. $$\dfrac{{^{10}\mathrm C_4}\,({^{4}\mathrm C_1})^4}{^{40}\mathrm C_4} = \dfrac{40\cdot 36\cdot 32\cdot 28}{40\cdot 39\cdot 38\cdot 37}$$
What is the probability that we draw at least three hearts.
For 2, I think something like 1 - Pr(choosing exactly 2 hearts) could work, but I don't know where to go from here.
That is not the correct complement.
The event of interest is drawing 3 or 4 from the four hearts (with the remainder of cards drawn being other suits). The complement of this is drawing 0, 1, or 2 from the four hearts.
Thus using the rule of complements is contra-indicated -- as calculating the probability for the complement would take more effort. (However, it will still get you the answer).
And you just use the rule of addition: the probability for the union of disjoint events is the sum of the probabilities for those events. So add the probability for drawing three hearts to the probability for drawing four hearts.
$$\dfrac{{^4\mathrm C_3}\,{^{36}\mathrm C_1}+{^4\mathrm C_4}\,{^{36}\mathrm C_0}}{^{40}\mathrm C_4}=1-\dfrac{{^4\mathrm C_0}\,{^{36}\mathrm C_4}+{^4\mathrm C_1}\,{^{36}\mathrm C_3}+{^4\mathrm C_2}\,{^{36}\mathrm C_2}}{^{40}\mathrm C_4}$$