In a deck of 52 cards, how many ways can you draw 2 cards without replacement? Options: a) 102, b) 2652, c) 132, d) 263.

We are selecting 2 cards from 52 without replacement. The phrasing "how many ways can you draw 2 cards" can be interpreted in two ways:

• If the order of drawing the cards does not matter (e.g., drawing King of Hearts then Queen of Spades is the same as drawing Queen of Spades then King of Hearts), it's a combination problem.
• If the order of drawing the cards does matter, it's a permutation problem.

• Number of combinations $C(n, k) = \frac{n!}{k!(n-k)!}$: $$C(52, 2) = \frac{52!}{2!(52-2)!} = \frac{52!}{2!50!} = \frac{52 \times 51}{2 \times 1} = 26 \times 51 = 1326$$
• Number of permutations $P(n, k) = \frac{n!}{(n-k)!}$: $$P(52, 2) = \frac{52!}{(52-2)!} = \frac{52!}{50!} = 52 \times 51 = 2652$$

The options are a) 102, b) 2652, c) 132, d) 263.
The combination result (1326) is not among the options.
The permutation result (2652) matches option b).
Given that 2652 is an option