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 calculate the number of different hands of cards possible, we need to consider the concept of combinations, as the order of the cards does not matter.
In this case, we want to choose 4 cards out of a deck of 44 cards. The number of combinations can be calculated using the formula for combinations, which is given by:
C(n, r) = n! / (r!(n-r)!),
where C(n, r) represents the number of combinations of choosing r items from a set of n distinct items.
Substituting the values into the formula, we have:
C(44, 4) = 44! / (4!(44-4)!)
= 44! / (4!40!)
= (44 x 43 x 42 x 41) / (4 x 3 x 2 x 1)
= 357,860.
Therefore, there are 357,860 different hands of 4 cards that can be dealt from a deck of 44 cards.
To calculate the number of different hands of cards possible, we need to use the concept of combinations.
In this case, we want to know how many combinations of 4 cards can be chosen from a deck of 44 cards. The order of the cards does not matter since it is a standard deck of cards.
The formula to calculate combinations is:
nCr = n! / (r! * (n-r)!)
Where:
n = total number of items in the set (44 cards in the deck)
r = number of items chosen from the set (4 cards in the hand)
! = factorial operator (the product of all positive integers from 1 to n)
Using this formula, we can calculate the number of different hands of cards possible as follows:
44! / (4! * (44-4)!)
Calculating this expression gives us:
44! / (4! * 40!)
Now, let's simplify the expression:
44! = 44 * 43 * 42 * 41 * 40!
4! = 4 * 3 * 2 * 1 (since 4! = 4 * 3 * 2 * 1)
40! = 40 * 39 * 38 * ... * 3 * 2 * 1 (simplified)
By canceling out the common terms between the numerator and denominator, we get:
= (44 * 43 * 42 * 41 * 40!) / (4 * 3 * 2 * 1 * 40!)
Notice that the (40!) terms in the numerator and denominator cancel out, leaving us with:
= (44 * 43 * 42 * 41) / (4 * 3 * 2 * 1)
Now we can simplify further:
= (44 * 43 * 42 * 41) / (24)
After performing the calculations, we find that there are 270,725 different hands of cards possible from a deck of 44 cards.