$$ (5.5,5.6,5.7) $$ 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 52 ways that three of the same card can be selected from the deck. (Type a whole number.) c) The remaining card must be different from the 3 chosen. After selecting the three of a kind, there are 12 different ranks of cards remaining in the deck that can be chosen. Of the 12 ranks remaining, the player chooses 1 of them and then selects one of the 4 cards in the chosen rank. How many ways can the player select the remaining card? There are 48 ways the player can select the remaining card. (Type a whole number.) d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and one card that is different? The probability of obtaining three of a kind from 4 cards dealt is (Round to four decimal places as needed.)

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To determine the probability of being dealt three of a kind from a standard 52-card deck, we will follow the steps outlined in the questions a through d.

### a) Total ways to select 4 cards from a 52-card deck
The number of ways to select 4 cards from a 52-card deck is given as 270,725.

### b) Ways to select three of the same card
Each rank has 4 cards (e.g., 4 aces, 4 kings). The number of ways to select three of the same card from the deck is 52, as there are 52 different ranks.

### c) Ways to select the remaining card
After selecting three of a kind, there are 12 different ranks remaining. The player can choose 1 of these ranks and then select one of the 4 cards in that rank. Therefore, the number of ways to select the remaining card is calculated as follows:

$$
\text{Ways to select remaining card} = 12 \text{ ranks} \times 4 \text{ cards} = 48
$$

### d) Probability of obtaining three of a kind
To find the probability of being dealt three of a kind, we can use the General Multiplication Rule. The total number of ways to get three of a kind and one different card is the product of the number of ways to select the three of a kind and the number of ways to select the remaining card.

1. **Ways to select three of a kind**: 52
2. **Ways to select the remaining card**: 48

Thus, the total number of favorable outcomes for three of a kind is:

$$
\text{Total favorable outcomes} = 52 \times 48
$$

Now, we can calculate the probability:

$$
\text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways to select 4 cards}} = \frac{52 \times 48}{270725}
$$

Let's calculate this probability.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{52\times 48}{270725}$$
- step1: Multiply the numbers:
  $$\frac{2496}{270725}$$
- step2: Reduce the fraction:
  $$\frac{192}{20825}$$
The total number of favorable outcomes for three of a kind is $$ 52 \times 48 = 2496 $$.

Now, the probability of obtaining three of a kind from 4 cards dealt is:

$$
\text{Probability} = \frac{2496}{270725} \approx 0.00922
$$

Rounding to four decimal places, the probability is approximately $$ 0.0092 $$.
The probability of being dealt three of a kind is approximately 0.0092.

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