Partition into aces and others. Over many 5 card hands, about 3.99% contain exactly 2 aces. \[
\frac{\binom{4}{2}\binom{48}{3}}{\binom{52}{5}}
\]
Partition into hearts and others. Over many 5 card hands, about 27.43% contain exactly 2 hearts. \[
\frac{\binom{13}{2}\binom{39}{3}}{\binom{52}{5}}
\]
Partition into ace of hearts, the 3 other aces, the 12 other hearts, and the 36 other cards. Over many 5 card hands, about 0.87% contain the ace of hearts, exactly one more ace, and exactly one more heart. \[
\frac{\binom{1}{1}\binom{3}{1}\binom{12}{1}\binom{36}{2}}{\binom{52}{5}}
\]
You need to choose 2 cards from 1 face, and one from the others. There are 4 possibilities for the face value that gets 2 cards. Over many 5 card hands, about 26.37% contains at least one card from each of the four suits. \[
4\frac{\binom{13}{2}\binom{13}{1}\binom{13}{1}\binom{13}{1}}{\binom{52}{5}}
\]
There are 13 choices for the face value of the four-of-a-kind, and 48 choices for the other card. Over many 5 card hands, about 0.02% contain a four-of-a-kind. \[
\frac{(13)(48)}{\binom{52}{5}}
\]
You deal more than 4 cards only if the first 4 cards are not hearts: 0.30382 \[
\left(\frac{39}{52}\right)\left(\frac{38}{51}\right)\left(\frac{37}{50}\right)\left(\frac{36}{49}\right)
\]