Answer :
Answer:
Step-by-step explanation:
Final answer:
The number of choices for the volunteer group is calculated using the formula for combinations. The percentage of choices where Pennsylvanians and Marylanders are equally represented can be found by dividing the number of ways to choose 3 volunteers from each state by the total number of choices.
Explanation:
To determine the number of choices for the volunteer group, we need to calculate the number of ways to choose 6 volunteers from a group of 38 students (20 from Pennsylvania and 18 from Maryland). This can be done using the formula for combinations:
nCr = n! / (r!(n-r)!)
Plugging in the values, we have:
C(38, 6) = 38! / (6!(38-6)!) = 38! / (6!32!)
Calculating this expression gives us the total number of choices for the volunteer group.
To calculate the percentage of choices where Pennsylvanians and Marylanders are equally represented, we need to find the number of ways to choose 3 volunteers from each state. This can be done using the same formula for combinations:
C(20, 3) * C(18, 3)
Dividing this by the total number of choices gives us the percentage of choices where Pennsylvanians and Marylanders are equally represented.
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Final answer:
There are over 2 million possible groups of volunteers. However, the calculated number of combinations where Pennsylvanians and Marylanders are equally represented exceeds the total number of combinations, indicating a possible error during calculations.
Explanation:
This is a combinatorial problem. We can calculate the number of groups we can form using the combination formula, [tex]C(n, k) = n! / (k!(n-k)!)[/tex], where n is the total number of students and k is the number of students we are choosing.
1. For the first part of the question, we have a total of 38 students (20 from Pennsylvania and 18 from Maryland) and we are looking for groups of 6. Using the combination formula, we get C(38, 6) = 2,760,681 possible groups.
2. For groups where Pennsylvanians and Marylanders are equally represented, we would need 3 students from each state. We calculate this separately for each state then multiply the results together. So, we have C(20, 3) for Pennsylvania and C(18, 3) for Maryland. We get 6,840 groups for Pennsylvania and 816 groups for Maryland. Multiplying these together we get 5,581,440. So, the desired groups form a fraction of [tex]5,581,440 / 2,760,681 = 2.02[/tex] which is more than 100%, indicating a miscalculation. There seem to be more combinations with an equal representation than total possible volunteer groups, which is not possible. Please check your numbers again.
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