Two cards are drawn from the pack of cards at random. What is the probability
that it will be a Diamond and Heart and King & Queen?


To solve this problem, we need to compute the probability of drawing two cards from a standard deck of 52 cards such that:

- One card is a **Diamond** and the other is a **Heart**.
- One card is a **King** and the other is a **Queen**.

### Step 1: Number of favorable outcomes
There are two scenarios that fulfill the given conditions:
1. The **King** is a Diamond and the **Queen** is a Heart.
2. The **Queen** is a Diamond and the **King** is a Heart.

Each of these cases represents a distinct favorable outcome.

- For scenario 1, the King of Diamonds and the Queen of Hearts is one valid outcome.
- For scenario 2, the Queen of Diamonds and the King of Hearts is another valid outcome.

Therefore, there are **2 favorable outcomes**.

### Step 2: Total number of possible outcomes
Since two cards are drawn at random from a deck of 52 cards, the total number of ways to choose 2 cards from 52 is given by the combination formula \( \binom{52}{2} \), which is:

\[
\binom{52}{2} = \frac{52 \times 51}{2} = 1326
\]

### Step 3: Probability calculation
The probability of drawing a Diamond and Heart that are a King and Queen is:

\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{1326}
\]

Simplifying the fraction:

\[
P = \frac{1}{663}
\]

### Final Answer:
The probability of drawing a Diamond and Heart that are a King and Queen is \( \frac{1}{663} \).

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Would you like more details or further clarification?

### Follow-up questions:
1. How many total cards are in each suit of a standard deck?
2. What is the probability of drawing two cards of the same suit?
3. How does the combination formula work for calculating probabilities?
4. How would the probability change if the deck had jokers included?
5. What is the probability of drawing any King and any Queen from a deck?

**Tip:** When calculating probabilities, always ensure you clearly distinguish between favorable outcomes and total possible outcomes.

Two cards are drawn from the pack of cards at random. What is the probability that it will be a Diamond and Heart and King & Queen?

Learn how to calculate the probability of drawing two specific cards from a deck: a Diamond, a Heart, a King, and a Queen. Using combinatorics and probability, this problem explores favorable outcomes, combination formulas, and total outcomes. Suitable for high school students in Grades 9-12.

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