If you draw five cards at random from a standard deck of 52 cards, what is the probability that there are at least 4 distinct characters (letters or numbers)? Report answer to 3 decimal places.

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

Explore the probability of drawing at least 4 distinct ranks from a standard deck of 52 cards. This detailed solution walks you through combinatorics and probability concepts, utilizing combination formulas to find the exact probability. Ideal for advanced high school students, Grades 11-12.

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