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A hand consists of 4 cards from a well-shuffled deck of 52 cards
a. Find the total number of possible 4-card poker hands
b. A diamond flush is a 4 -card hand consisting of all diamond cards. Find the number of possible diamond flushes.
c. Find the probability of being dealt a diamond flush.
a. There are a total of
◻ poker hands.

A hand consists of $4$ cards from a well-shuffled deck of $52$ cards $\newline$ a. Find the total number of possible $4$ -card poker hands $\newline$ b. A diamond flush is a $4$ -card hand consisting of all diamond cards. Find the number of possible diamond flushes. $\newline$ c. Find the probability of being dealt a diamond flush. $\newline$ a. There are a total of $\square$ poker hands.

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Find probabilities using the addition rule

Full solution

Q. A hand consists of

4

cards from a well-shuffled deck of

52

cards

\newline

a. Find the total number of possible

4

-card poker hands

\newline

b. A diamond flush is a

4

-card hand consisting of all diamond cards. Find the number of possible diamond flushes.

\newline

c. Find the probability of being dealt a diamond flush.

\newline

a. There are a total of

\square

poker hands.

Use Combination Formula: To find the total number of possible $4$ -card poker hands from a deck of $52$ cards, we use the combination formula which is given by $C(n, k) = \frac{n!}{k! \cdot (n - k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, and “ $!$ ” denotes factorial. In this case, $n = 52$ and $k = 4$ .
Calculate Total Number: Calculating the total number of possible $4$ -card hands using the combination formula: $C(52, 4) = \frac{52!}{4! \times (52 - 4)!} = \frac{52!}{4! \times 48!}$ .
Simplify Factorials: Simplifying the factorials by canceling out the common terms: $\frac{52!}{(4! \times 48!)} = \frac{(52 \times 51 \times 50 \times 49)}{(4 \times 3 \times 2 \times 1)}$ .
Perform Multiplication and Division: Performing the multiplication and division: $(52 \times 51 \times 50 \times 49) / (4 \times 3 \times 2 \times 1) = 270725$ .

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