a game of cards is played with a hand that consists of 4 cards dealt from a deck of 44 cards how many different hands of cards are possible
To determine the number of different hands of 4 cards that are possible from a deck of 44 cards, we can use the combination formula.
The formula for combinations is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of items to choose from and r is the number of items to be chosen.
In this case, we have a deck of 44 cards, and we want to choose 4 cards for the hand. Plugging these values into the formula:
C(44, 4) = 44! / (4!(44-4)!)
= 44! / (4!40!)
Calculating this expression, it simplifies to:
C(44, 4) = (44 * 43 * 42 * 41 * 40!) / (4! * 40!)
= (44 * 43 * 42 * 41) / (4 * 3 * 2 * 1)
= (543,312) / (24)
= 22,638
Therefore, there are 22,638 different hands of cards possible from a deck of 44 cards.
To calculate the number of different hands of cards possible in a game played with 4 cards from a deck of 44 cards, you can use the concept of combinations.
The formula to calculate combinations is written as:
nCr = n! / (r! * (n - r)!)
Where n represents the total number of items to choose from, and r represents the number of items to be chosen.
In this case, there are 44 cards in the deck, and we want to choose 4 cards for a hand. So, applying the formula:
44C4 = 44! / (4! * (44 - 4)!)
Now let's perform the calculations step-by-step:
1. Calculate 44! (44 factorial):
- 44! = 44 × 43 × 42 × ... × 2 × 1
2. Calculate 4! (4 factorial):
- 4! = 4 × 3 × 2 × 1
3. Calculate (44 - 4)! (40 factorial):
- (44 - 4)! = 40! = 40 × 39 × 38 × ... × 2 × 1
4. Substitute the values into the formula:
- 44C4 = 44! / (4! * (44 - 4)!)
- 44C4 = (44 × 43 × 42 × ... × 2 × 1) / ((4 × 3 × 2 × 1) * (40 × 39 × 38 × ... × 2 × 1))
5. Simplify the expression:
- 44C4 = (44 × 43 × 42 × ... × 2 × 1) / (4 × 3 × 2 × 1 × (40 × 39 × 38 × ... × 2 × 1))
Using a calculator, the result of the calculation is 91,390.
Therefore, there are 91,390 different hands of cards possible in this game.