A player is dealt 4 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering questions a through d.
a) How many ways can 4 cards be selected from a 52 -card deck?
There are 270,725 ways that 4 cards can be selected from a 52 -card deck.
(Type a whole number.)
b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck?
There are ways that three of the same card can be selected from the deck.
(Type a whole number.)

Question

A player is dealt 4 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering questions a through d.
a) How many ways can 4 cards be selected from a 52 -card deck?
There are 270,725 ways that 4 cards can be selected from a 52 -card deck.
(Type a whole number.)
b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck?
There are

ways that three of the same card can be selected from the deck.
(Type a whole number.)

UpStudy AI · Accepted Answer

To solve part b, we need to determine how many ways three of the same card can be selected from a standard 52-card deck.

1. **Choosing the Rank**: There are 13 different ranks (2 through Ace) in a deck. We can choose any one of these ranks to have three of a kind.

2. **Choosing the Suits**: For the chosen rank, there are 4 cards (one for each suit). We need to select 3 out of these 4 cards.

The number of ways to choose the rank is $$ 13 $$, and the number of ways to choose 3 suits from 4 is given by the combination formula $$ \binom{n}{k} $$, where $$ n $$ is the total number of items to choose from, and $$ k $$ is the number of items to choose.

Thus, the number of ways to choose 3 suits from 4 is:

$$
\binom{4}{3} = 4
$$

Now, we can calculate the total number of ways to select three of the same card:

$$
\text{Total ways} = \text{(ways to choose rank)} \times \text{(ways to choose suits)} = 13 \times 4
$$

Now, let's calculate this.
Calculate the value by following steps:
- step0: Calculate:
  $$13\times 4$$
- step1: Multiply the numbers:
  $$52$$
There are $$ 52 $$ ways that three of the same card can be selected from the deck.

Now, let's summarize the answers:

a) There are 270,725 ways that 4 cards can be selected from a 52-card deck.

b) There are $$ 52 $$ ways that three of the same card can be selected from the deck.
a) There are 270,725 ways to select 4 cards from a 52-card deck. b) There are 52 ways to select three of the same card from the deck.

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