Question

Using a 52 card deck, how many 5 card hands have either 5 hearts or 4 hearts and 1 club

Answer 1

Answer:

10,582

Step-by-step explanation:

We can choose 5 cards from 52 card deck in

      $n = \binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = \frac{\cancel{47!} \cdot 48 \cdot 49 \cdot 50 \cdot 51 }{5! \cancel{47!}} = \frac{ 48 \cdot 49 \cdot 50 \cdot 51 }{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} = 2 \; 598 \; 960$

ways.

Now, let's calculate the number of ways we can choose 5 hearts. We know that in a 52 card deck, we have 13 hearts. Therefore, the number of ways to choose 5 hearts is

       $n_1 = \binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13!}{5!8!} = \frac{8! \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13}{5!8!} = \frac{9 \cdot 10 \cdot 11 \cdot 12 \cdot 13}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} = 1287$

Similarly, number of ways to choose 4 hearts equals $\binom{13}{4}$ and number of ways to choose 1 club equals $\binom{13}{1}$, since there are also 13 clubs in the deck.

Therefore, the number of ways of choosing 4 hearts and 1 club equals

                                   $n_2 = \binom{13}{4} \cdot \binom{13}{1} = 9295$

The probability of this event is calculated as

           $P(A) = \frac{\text{total number of ways to choose 5 hearts or 4 hearts and a club}}{\text{total number of ways to choose 5 cards from a deck of 52 cards}}$

Therefore

                     $P(A) = \frac{n_1+n_2}{n} = \frac{1287+9295}{2598960} =0.0040716 \approx 0.0041$