To find the probability that there are at least 4 distinct characters when drawing 5 cards from a standard deck of 52, we can approach the problem step-by-step.

Step 1: Understanding the deck and possibilities

A standard deck has:

• 13 ranks (A, 2, 3, ..., 10, J, Q, K)
• 4 suits for each rank (spades, hearts, diamonds, clubs)

Thus, there are 13 distinct "characters" (ranks), and we are tasked with finding the probability of having at least 4 distinct ranks when drawing 5 cards.

Step 2: Total possible outcomes

The total number of ways to choose 5 cards from 52 is given by the combination formula $\binom{52}{5}$:

$\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960$

Step 3: Unfavorable outcomes (less than 4 distinct ranks)

We now count the number of outcomes where there are fewer than 4 distinct ranks (i.e., 3 or fewer distinct ranks).

3 distinct ranks:

To have exactly 3 distinct ranks in 5 cards:

• Choose 3 ranks from 13: $\binom{13}{3}$
• Distribute the 5 cards among these 3 ranks. The possible distributions are either (3, 2, 0) or (3, 1, 1) or (2, 2, 1):
  • (3, 2): One rank appears 3 times, and another appears 2 times: $\binom{4}{3} \times \binom{4}{2}$
  • Continuing combinations