Questions: If 5 cards are dealt from a standard deck of cards, how many different ways can two hearts and three non-hearts be dealt? There are five-card hands consisting of two hearts and three non-hearts. (Type a whole number.)

Solution Steps

To solve this problem, we need to calculate the number of ways to choose 2 hearts from the 13 available hearts in a standard deck, and then choose 3 non-heart cards from the remaining 39 cards (since there are 52 cards in total and 13 are hearts). We will use combinations to determine the number of ways to choose these cards.

To find the number of ways to choose 2 hearts from 13 available hearts, we use the combination formula:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

For our case:

\[ \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 \]

Next, we calculate the number of ways to choose 3 non-heart cards from the remaining 39 cards:

\[ \binom{39}{3} = \frac{39!}{3!(39-3)!} = \frac{39 \times 38 \times 37}{3 \times 2 \times 1} = 9139 \]

Now, we find the total number of different five-card hands consisting of 2 hearts and 3 non-hearts by multiplying the two results:

\[ \text{Total Ways} = \binom{13}{2} \times \binom{39}{3} = 78 \times 9139 = 712842 \]

Final Answer