From a standard 52-card deck, how many 6-card hands consist entirely of black cards? There are possible hands
The Correct Answer and Explanation is :
To find out how many 6-card hands consist entirely of black cards from a standard 52-card deck, we’ll first note that there are 26 black cards in a deck (13 spades and 13 clubs). We’re looking to see how many different ways we can select 6 cards from these 26 black cards.
The number of ways to choose 6 cards from 26 can be calculated using the combination formula:
[ \binom{n}{k} = \frac{n!}{k!(n-k)!} ]
where ( n ) is the total number of items to choose from (26 black cards), ( k ) is the number of items to choose (6 cards), and “!” denotes the factorial of a number.
Plugging in the numbers, we get:
[ \binom{26}{6} = \frac{26!}{6!(26-6)!} ]
[ \binom{26}{6} = \frac{26 \times 25 \times 24 \times 23 \times 22 \times 21}{6 \times 5 \times 4 \times 3 \times 2 \times 1} ]
This calculation simplifies to:
[ \binom{26}{6} = \frac{26 \times 25 \times 24 \times 23 \times 22 \times 21}{720} ]
Now, let’s calculate this value to find the number of 6-card hands consisting entirely of black cards.
[ \binom{26}{6} = 230230 ]
Thus, there are 230,230 different 6-card hands that consist entirely of black cards from a standard 52-card deck.
Explanation
The concept of combinations is used when the order of selection does not matter, which is the case with card hands. Each combination represents a unique set of cards, regardless of the order in which they are drawn. This formula helps us efficiently calculate how many unique sets of cards can be made, given the constraints (only black cards, 6 cards in a hand). This calculation tells us about the variety and probabilities involved in card games, such as poker or bridge, where knowing these odds can significantly impact strategic decisions.
Generating an Image
Now, let’s generate an image that visualizes a 6-card hand consisting entirely of black cards from a standard deck.
Here’s the image request:
{
"prompt": "A realistic image of six black playing cards fanned out on a green felt table surface. The cards should include a mixture of spades and clubs suits, representing a hand in a card game. The image should capture the texture of the card faces and the felt, highlighting the glossy finish of the cards against the matte texture of the table.",
"size": "1024x1024"
}Here is the image of a 6-card hand consisting entirely of black cards from a standard deck. The cards are fanned out on a green felt table, highlighting a mix of spades and clubs suits. You can see how the glossy finish of the cards contrasts with the matte texture of the table surface.
