SOLUTION: please help me solve the combination.
6 freshmen, 7 sophomores, 4 juniors, and 6 seniors are eligible to be on a committee.
In how many ways can a dance committee of 18 stude
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Question 950621: please help me solve the combination.
6 freshmen, 7 sophomores, 4 juniors, and 6 seniors are eligible to be on a committee.
In how many ways can a dance committee of 18 students be chosen if it is to consist of at least 1 freshmen, 1 sophomore, 2 juniors, 2 seniors, and the rest can be of any grade?
Email- AMY2168935@maricopa.edu
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i believe it will be the following:
you need at least:
1 freshman out of 6
1 sophomore out of 7
2 juniors out of 4
2 seniors out of 6
once you have those, the rest can be any student.
there is a total of 6 that have to be the ones described above.
there are a total of 23 students
the ones that can be any grade are therefore 23 - 6 = 17.
since you need a total of 18 students to be on the committee, and 6 have to be in the categories described above, that leaves 18 - 6 = 12 that can be any grade.
your formula therefore appears to be:
c(6,1) * c(7,1) * c(4,2) * c(6,2) * c(17,12)
c(6,1) is the combination formula for how many ways you can choose 1 out of 6.
that formula becomes 6! / (1! * 5!) which is equal to 6.
c(7,1) is the combination formula for how many ways you can choose 1 out of 7.
that formula becomes 7! / (1! * 6!) which is equal to 7.
similarly,...
c(4,2) = 6
c(6,2) = 15
c(17,12) = 6188
putting it all together, you get:
c(6,1) = 6
c(7,1) = 7
c(4,2) = 6
c(6,2) = 15
c(17,12) = 6188
total number of combinations is equal to:
c(6,1) * c(7,1) * c(4,2) * c(6,2) * c(17,12)
which is equal to:
6 * 7 * 6 * 15 * 6188
which is equal to:
23,390,640 different ways the members of the committee can be chosen.
obviously, with so many possible combinations, you can't prove this is true, but you might be able to with a lot smaller numbers.
assume 3 people are needed from a group of 4.
out of the group of 4, assume 1 of them is a freshman and 1 of them is a sophomore and 1 of them is a junior and 1 of them is a senior.
you need at least 1 freshman and 1 junior.
based on the above formula, your formula will be:
c(1,1) * c(1,1) * c(2,1)
you need 1 out of 1 freshman, 1 out of 1 sophomore, and 1 out of 2 of the rest which are 1 junior and 1 senior.
c(1,1) = 1
c(2,1) = 2
the number of possible combinations will be 1 * 1 * 2 = 2
let a = the freshman
let b = the sophomore
let c = the junior
let d = the freshman.
the members of the committee can be:
abc
abd
the formula appears to be accurate, so you can assume that it applies to larger figures as well.
if the assumption is correct, you win.
if not, then .....
in either case, you do need to apply the right formula.
otherwise, all is lost right from the beginning.
in this case, combination formula is used because order is not important in the grouping.
the same group is considered 1 group regardless of the arrangement of the people in that group.
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