Questions: If a coin is tossed 4 times, and then a standard six-sided die is rolled 2 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

If a coin is tossed 4 times, and then a standard six-sided die is rolled 2 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

Transcript text: If a coin is tossed 4 times, and then a standard six-sided die is rolled 2 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

Solution

Solution Steps

Step 1: Coin Tosses

For each toss of a coin, there are 2 possible outcomes. Therefore, if the coin is tossed 4 times, the total number of outcomes is $2^4 = 16$.

Step 2: Die Rolls

For each roll of a six-sided die, there are 6 possible outcomes. Therefore, if the die is rolled 2 times, the total number of outcomes is $6^2 = 36$.

Step 3: Card Draws

Drawing cards without replacement from a deck of 52 means the number of possible outcomes decreases with each draw. Therefore, the total number of outcomes for drawing 3 cards is given by the formula for permutations of 52 objects taken 3 at a time, which is $\frac{52!}{(52-3)!} = 132600$.

Step 4: Total Outcomes

To find the total number of different outcomes for the entire sequence of events, multiply the outcomes of each event together. This gives the formula: $Total\ Outcomes = 2^4 \times 6^2 \times \frac{52!}{(52-3)!} = 76377600$.

Final Answer:

The total number of different outcomes possible is 76377600.

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