Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12?

To determine the probability that both cards chosen from a standard 52-card deck are numbers (2 through 10) totaling to 12, we can follow these steps:

### Step 1: Identify the possible card pairs

First, let's list the pairs of number cards (2 through 10) that sum to 12:
- 2 and 10
- 3 and 9
- 4 and 8
- 5 and 7
- 6 and 6

### Step 2: Calculate the number of favorable outcomes

For each pair, we need to consider how many ways we can choose the cards:

- **(2, 10)**: There are 4 twos and 4 tens in the deck. The number of ways to choose one 2 and one 10 is \(4 \times 4 = 16\).
- **(3, 9)**: There are 4 threes and 4 nines. The number of ways to choose one 3 and one 9 is \(4 \times 4 = 16\).
- **(4, 8)**: There are 4 fours and 4 eights. The number of ways to choose one 4 and one 8 is \(4 \times 4 = 16\).
- **(5, 7)**: There are 4 fives and 4 sevens. The number of ways to choose one 5 and one 7 is \(4 \times 4 = 16\).
- **(6, 6)**: There are 4 sixes. The number of ways to choose two 6s is given by combinations \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6\).

Summing these, the total number of favorable outcomes is:
\[16 + 16 + 16 + 16 + 6 = 70\]

### Step 3: Calculate the total number of outcomes

The total number of ways to choose 2 cards out of 52 is given by the combinations \(\binom{52}{2}\):
\[
\binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326
\]

### Step 4: Compute the probability

The probability is the ratio of the number of favorable outcomes to the total number of outcomes:
\[
P(\text{sum to 12}) = \frac{70}{1326} = \frac{35}{663}
\]

Therefore, the probability that both cards are numbers (2 through 10) totaling to 12 is:
\[
\boxed{\frac{35}{663}}
\]

Would you like further details or have any questions?

Here are some related questions:

1. What is the probability of drawing two face cards in a row without replacement?
2. How many ways can you draw two aces from a deck?
3. What is the probability of drawing two cards of the same suit in a row?
4. How does the probability change if we draw cards with replacement?
5. What is the probability of drawing one red and one black card?
6. How many ways can you draw two cards that are both hearts?
7. What is the probability of drawing two cards that are both greater than 5?
8. How many ways can you draw a pair of jacks from the deck?

**Tip:** When dealing with card probabilities, always consider whether the problem involves drawing with or without replacement, as it significantly affects the total number of outcomes.

Explore the probability of drawing two number cards (2 through 10) from a standard 52-card deck that total to 12. Learn step-by-step calculations involving combinations, basic arithmetic, and probability concepts.

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