To find how many different outcomes are possible, let's break the problem into its three distinct parts:

1. Coin Tossed 3 Times:

Each coin toss has 2 possible outcomes (Heads or Tails), and since the coin is tossed 3 times, the number of possible outcomes is:

$2^3 = 8$

2. Die Rolled 3 Times:

Each roll of a standard six-sided die has 6 possible outcomes. Since the die is rolled 3 times, the number of possible outcomes is:

$6^3 = 216$

3. Five Cards Drawn Without Replacement:

From a standard deck of 52 cards, we are selecting 5 cards without replacement. The number of ways to choose 5 cards from 52 can be found using the combination formula:

$\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960$

Total Number of Different Outcomes:

The total number of different outcomes is the product of the outcomes from each stage:

$\text{Total outcomes} = 2^3 \times 6^3 \times \binom{52}{5}$

$\text{Total outcomes} = 8 \times 216 \times 2,598,960$

$\text{Total outcomes} = 4,494,110,080$

So, the total number of different outcomes is 4,494,110,080.

Would you like a more detailed explanation on any part?

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5 Related Questions:

1. How many outcomes are possible if the coin is tossed 4 times instead of 3?
2. What is the probability of drawing 5 cards of the same suit from a deck?
3. How would the total outcomes change if the die is rolled 4 times?
4. What if we draw 4 cards instead of 5 from the deck of 52? How many outcomes are possible?
5. Can the combination formula be applied if the cards were drawn with replacement?

Tip:

When calculating combinations (like card draws), remember that order doesn't matter. For outcomes where order matters, you'd use permutations instead.