After removing all the face cards from a deck, four cards are to be drawn in total. First two cards are drawn simultaneously without replacement and the next two cards are then again drawr simultaneously. What would be the probability that the first two cards add up to 15 and the next two cards add up to 17?

In this problem, we're dealing with a modified deck of cards where all the face cards (Jack, Queen, and King) have been removed. Let's break down the scenario:

### 1. Understanding the Modified Deck:
- A standard deck has 52 cards. After removing the face cards (Jack, Queen, and King), each suit (hearts, diamonds, spades, clubs) will have 10 cards each: Ace, 2 through 10. 
- This means we are left with a total of \( 40 \) cards in the deck.

### 2. The Objective:
- You are to draw 4 cards in total: the first two drawn simultaneously without replacement and the next two also drawn simultaneously without replacement.
- The goal is:
  - The first two cards should add up to 15.
  - The next two cards should add up to 17.

### 3. Possible Pairs that Sum to 15:
Let’s first find all pairs of values in the modified deck that sum up to 15.

- \( 10 + 5 = 15 \)
- \( 9 + 6 = 15 \)
- \( 8 + 7 = 15 \)

So the possible pairs that sum to 15 are \( (10, 5), (9, 6), (8, 7) \).

### 4. Possible Pairs that Sum to 17:
Now, let's find all pairs that sum to 17.

- \( 10 + 7 = 17 \)
- \( 9 + 8 = 17 \)

So the possible pairs that sum to 17 are \( (10, 7), (9, 8) \).

### 5. Total Possible Ways to Draw Cards:
Since there are 40 cards, the number of ways to choose 4 cards out of these 40 is calculated as:
\[
\binom{40}{4} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,390
\]
So there are 91,390 possible ways to draw any 4 cards from the deck.

### 6. Favorable Outcomes:
We now need to calculate the number of favorable outcomes where the first two cards sum to 15 and the next two sum to 17.

#### Case 1: First Pair Adds to 15
The valid pairs for a sum of 15 are \( (10, 5), (9, 6), (8, 7) \). Each pair can be chosen in \( 4 \times 4 = 16 \) ways (since there are 4 suits for each number). So, there are:
\[
3 \times 16 = 48 \text{ ways to choose two cards that sum to 15}.
\]

#### Case 2: Second Pair Adds to 17
The valid pairs for a sum of 17 are \( (10, 7), (9, 8) \). Each pair can be chosen in \( 4 \times 4 = 16 \) ways. So, there are:
\[
2 \times 16 = 32 \text{ ways to choose two cards that sum to 17}.
\]

### 7. Total Number of Favorable Outcomes:
The number of favorable outcomes is the product of the favorable outcomes for the two draws:
\[
48 \times 32 = 1,536.
\]

### 8. Probability Calculation:
Finally, the probability is the ratio of favorable outcomes to total outcomes:
\[
P = \frac{1,536}{91,390} \approx 0.0168.
\]

Thus, the probability that the first two cards sum to 15 and the next two sum to 17 is approximately \( 0.0168 \) or \( 1.68\% \).

---

Would you like further explanation on any part of the solution? 

Here are some related questions for further exploration:

1. What would be the probability if we also include face cards?
2. How would the probability change if we drew cards one at a time instead of simultaneously?
3. What are the odds of drawing a pair that sums to a specific number in one draw?
4. How does the probability change if the deck is reshuffled after each draw?
5. What would the probability be if we draw 5 cards instead of 4?

**Tip**: Understanding combinations (\( \binom{n}{r} \)) and how to apply them is crucial for solving problems involving selections and probabilities in card games.

After removing all the face cards from a deck, four cards are to be drawn in total. First two cards are drawn simultaneously without replacement and the next two cards are then again drawn simultaneously. What would be the probability that the first two cards add up to 15 and the next two cards add up to 17?

Discover how to calculate the probability of drawing four cards from a deck where face cards are removed, with two cards summing to 15 and two cards summing to 17. Learn the combinatorial approach and probability formulas. Suitable for Grades 10-12 students.

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