Let's solve each part of the question one by one.

1. Probability that both cards are of different colors:

In a standard deck of 52 cards:

• There are 26 red cards (hearts and diamonds).
• There are 26 black cards (spades and clubs).

We want to find the probability that one card is red and the other is black.

Total number of ways to choose 2 cards from 52 cards:

$\binom{52}{2} = \frac{52 \times 51}{2} = 1326$

Favorable outcomes (one red and one black):

• Select one red card from 26 red cards: $\binom{26}{1} = 26$
• Select one black card from 26 black cards: $\binom{26}{1} = 26$

Thus, the total number of favorable outcomes: $26 \times 26 = 676$

Probability:

$\text{Probability} = \frac{676}{1326} = \frac{338}{663}$

2. Probability that both cards are face cards:

There are 12 face cards in total (4 kings, 4 queens, and 4 jacks).

Total number of ways to choose 2 cards from 52 cards:

$\binom{52}{2} = 1326$

Favorable outcomes (both cards are face cards):

• Select 2 face cards from 12 face cards: $\binom{12}{2} = \frac{12 \times 11}{2} = 66$

Probability:

$\text{Probability} = \frac{66}{1326} = \frac{11}{221}$

3. Probability that one of the two cards is a king:

There are 4 kings in the deck.

Total number of ways to choose 2 cards from 52 cards:

$\binom{52}{2} = 1326$

Favorable outcomes (one card is a king):

• Choose 1 king from 4 kings: $\binom{4}{1} = 4$
• Choose 1 non-king card from the remaining 48 cards: $\binom{48}{1} = 48$

Thus, the total number of favorable outcomes: $4 \times 48 = 192$

Probability:

$\text{Probability} = \frac{192}{1326} = \frac{32}{221}$

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Summary of results:

1. Probability that both cards are of different colors: $\frac{338}{663}$
2. Probability that both cards are face cards: $\frac{11}{221}$
3. Probability that one of the two cards is a king: $\frac{32}{221}$

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Do you want details on any of these steps or have any questions?

Here are 5 related questions for further exploration:

1. What is the probability that both cards drawn are red?
2. What is the probability of drawing two aces from the deck?
3. What is the probability that neither of the two cards is a face card?
4. How does the probability change if 3 cards are drawn instead of 2?
5. What is the probability of drawing at least one face card in two draws?

Tip: When calculating probabilities involving combinations, always break down the problem into small steps and compute each part methodically.