Question

five cards are chosen at random from an ordinary deck to form a hand in poker. in how many ways is it possible to get the following results? complete parts (a) through (e) below.(a) how many ways is it possible to choose 4 sixes?there are 48 ways.(b) how many ways is it possible to choose no hearts?there are 575757 ways.(c) how many ways is it possible to choose exactly 3 hearts?there are 211926 ways.(d) how many ways is it possible to choose at least 3 hearts?there are □ ways

Explanation:

Step1: Define "at least 3 hearts"

This includes 3 hearts + 2 non-hearts, 4 hearts + 1 non-heart, and 5 hearts.

Step2: Calculate 3 hearts + 2 non-hearts

Choose 3 from 13 hearts, 2 from 39 non-hearts:
$\binom{13}{3} \times \binom{39}{2} = \frac{13!}{3!(13-3)!} \times \frac{39!}{2!(39-2)!} = 286 \times 741 = 211926$

Step3: Calculate 4 hearts + 1 non-heart

Choose 4 from 13 hearts, 1 from 39 non-hearts:
$\binom{13}{4} \times \binom{39}{1} = \frac{13!}{4!(13-4)!} \times 39 = 715 \times 39 = 27885$

Step4: Calculate 5 hearts

Choose 5 from 13 hearts:
$\binom{13}{5} = \frac{13!}{5!(13-5)!} = 1287$

Step5: Sum all valid combinations

Add the three results together:
$211926 + 27885 + 1287$

Answer:

241098