Question

How many ways is it possible to choose "at least" 2 spades from a 5 card hand?

Answer 1

To find the number of ways to choose at least 2 spades from a 5 card hand, we can break it down into two cases: choosing exactly 2 spades and 3 non-spades, and choosing 3 or more spades and 2 or fewer non-spades. By calculating the combinations for each case, we find that there are 718,770 ways to choose at least 2 spades.

Step-by-step explanation:

To find the number of ways to choose at least 2 spades from a 5 card hand, we can break it down into two cases:

1. Choosing exactly 2 spades and 3 non-spades.
2. Choosing 3 or more spades and 2 or fewer non-spades.

For case 1, we can choose the 2 spades from the 13 spades in the deck in 13C2 = 78 ways. Then we can choose the 3 non-spades from the 39 non-spades in the deck in 39C3 = 9139 ways. So, there are 78 * 9139 = 716,482 ways to choose exactly 2 spades and 3 non-spades.

For case 2, we can choose 3 or more spades from the 13 spades in the deck in the following ways: 3 spades in 13C3 = 286 ways, 4 spades in 13C4 = 715 ways, and 5 spades in 13C5 = 1287 ways. The remaining non-spades can be chosen in 39C2 = 741 ways. So, there are a total of 286 + 715 + 1287 = 2288 ways to choose 3 or more spades and 2 or fewer non-spades.

Therefore, the total number of ways to choose at least 2 spades from a 5 card hand is 716,482 + 2288 = 718,770.

Answer 2

Depends on the hand, although it’s safe to assume that this is the standard, so 5 x 52 divided by 2 should give us the mean, then use the comparison property of probability to divide that by 2, you get 89 possible ways.