To solve this problem, we need to calculate the probability of being dealt exactly 4 face cards in an 8-card poker hand from a standard deck of 52 cards.

Step 1: Determine the total number of possible hands

The number of ways to choose 8 cards from a deck of 52 cards is given by the combination formula:

$\binom{52}{8} = \frac{52!}{8!(52-8)!} = \binom{52}{8}$

Step 2: Determine the number of successful hands (hands with exactly 4 face cards)

There are 12 face cards in a deck (4 Jacks, 4 Queens, and 4 Kings). We want to choose exactly 4 of these 12 face cards. The number of ways to choose 4 face cards from 12 is:

$\binom{12}{4} = \frac{12!}{4!(12-4)!} = \binom{12}{4}$

The remaining 4 cards must come from the 40 non-face cards in the deck. The number of ways to choose 4 non-face cards from the 40 non-face cards is:

$\binom{40}{4} = \frac{40!}{4!(40-4)!} = \binom{40}{4}$

Step 3: Calculate the probability

The probability of getting exactly 4 face cards is the ratio of the number of successful hands to the total number of possible hands. This can be expressed as:

$\text{Probability} = \frac{\binom{12}{4} \cdot \binom{40}{4}}{\binom{52}{8}}$

Now, let's calculate the combinations and simplify.

Calculations:

$\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$ $\binom{40}{4} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,390$ $\binom{52}{8} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \times 45}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 752,875,680$

Now, calculate the probability:

$\text{Probability} = \frac{495 \times 91,390}{752,875,680} = \frac{45,276,050}{752,875,680}$

Simplifying the fraction:

[ \text{Probability} = \frac{226,38025}{376,437,840} = \frac{22635}{18821892} \