As President of Spirit Club, Carson organized a "Day of Decades" fundraiser where students could pay a fixed amount to dress up as their favorite decade. Of the 56 students who participated, 10 of them dressed up as the '70s and 36 dressed up as the '90s.
If Carson randomly chose 4 of the participants to take pictures of for the yearbook, what is the probability that 2 of the chosen students dressed up as the '70s and 2 dressed up as the '90s?
Write your answer as a decimal rounded to four decimal places.

We are tasked with finding the probability that, when Carson randomly selects 4 students out of the 56 who participated in the fundraiser, exactly 2 dressed up as the '70s and 2 dressed up as the '90s.

### Step 1: Total number of students and selections
- Total number of participants = 56
- Students dressed as the '70s = 10
- Students dressed as the '90s = 36

### Step 2: Total number of ways to select 4 students from 56
The total number of ways to choose 4 students from 56 is given by the combination formula:

\[
\binom{56}{4} = \frac{56 \times 55 \times 54 \times 53}{4 \times 3 \times 2 \times 1}
\]

### Step 3: Number of ways to select 2 students from the '70s and 2 from the '90s
To select exactly 2 students from the 10 who dressed as the '70s, the number of ways is:

\[
\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
\]

To select exactly 2 students from the 36 who dressed as the '90s, the number of ways is:

\[
\binom{36}{2} = \frac{36 \times 35}{2 \times 1} = 630
\]

Thus, the number of favorable outcomes is the product of these two combinations:

\[
\binom{10}{2} \times \binom{36}{2} = 45 \times 630 = 28350
\]

### Step 4: Calculating the probability
The probability of selecting exactly 2 students from the '70s and 2 from the '90s is the ratio of favorable outcomes to total outcomes:

\[
P = \frac{\binom{10}{2} \times \binom{36}{2}}{\binom{56}{4}} = \frac{28350}{367290}
\]

Now, calculating the decimal form:

\[
P = \frac{28350}{367290} \approx 0.0772
\]

### Final Answer:
The probability is approximately \( \boxed{0.0772} \).

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. What if Carson chose 5 participants instead? How would the probability change?
2. How many students would need to dress as the '80s for the probability of selecting 1 from each decade to be calculated?
3. If Carson selected only 3 students, what is the probability that all dressed as the '90s?
4. What is the probability that none of the selected students dressed as the '70s?
5. How does the probability change if the total number of participants is increased?

**Tip:** Always double-check if combinations are the best approach, especially when dealing with selection problems involving multiple categories.

As President of Spirit Club, Carson organized a 'Day of Decades' fundraiser where students could pay a fixed amount to dress up as their favorite decade. Of the 56 students who participated, 10 of them dressed up as the '70s and 36 dressed up as the '90s. If Carson randomly chose 4 of the participants to take pictures of for the yearbook, what is the probability that 2 of the chosen students dressed up as the '70s and 2 dressed up as the '90s? Write your answer as a decimal rounded to four decimal places.

Learn how to calculate the probability of selecting 2 students dressed as the '70s and 2 dressed as the '90s from a group of 56 participants in a 'Day of Decades' fundraiser. This problem involves key combinatorics and probability concepts. Suitable for students in Grades 9-12.

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