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In a high school debating team consisting of 2 freshmen, 2 sophomores

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Bunuel
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In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink] New post   12 Nov 2021, 07:15
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In a high school debating team consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors, two students are selected to represent the school at the state debating championship. The rules stipulate that the representatives must be from different grades, but otherwise the 2 representatives are to be chosen by lottery. What is the probability that the students selected will consist one freshman and one sophomore?

(A) 1/16
(B) 1/8
(C) 1/7
(D) 1/6
(E) 1/4
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chetan2u
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Re: In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink] New post   05 Jan 2022, 18:38
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Bunuel wrote:
In a high school debating team consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors, two students are selected to represent the school at the state debating championship. The rules stipulate that the representatives must be from different grades, but otherwise the 2 representatives are to be chosen by lottery. What is the probability that the students selected will consist one freshman and one sophomore?

(A) 1/16
(B) 1/8
(C) 1/7
(D) 1/6
(E) 1/4



Arvind123

Ways to select one freshman(F) and one sophomore(S)
Both F and S can be chosen in 2C1 ways, so 2C1*2C1 or 2*2 or 4 ways

Ways to select 2 people such that both are from different group.
There are total 8 people, so we can select 2 out of them in \(\frac{8!}{6!2!}\) or 28 ways
But these 28 ways consist of 4 ways where both F or both S or both juniors or both seniors are together => 28-4 or 24 ways.


Thus, P = \(\frac{4}{24}=\frac{1}{6}\)
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KarishmaB
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Re: In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink] New post   05 Jan 2022, 22:32
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Bunuel wrote:
In a high school debating team consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors, two students are selected to represent the school at the state debating championship. The rules stipulate that the representatives must be from different grades, but otherwise the 2 representatives are to be chosen by lottery. What is the probability that the students selected will consist one freshman and one sophomore?

(A) 1/16
(B) 1/8
(C) 1/7
(D) 1/6
(E) 1/4


Total number of acceptable ways: 8C2 (select any 2 out of 8) - 4 (4 cases in which both selected are from same grade are unacceptable) = 24
Total number of favourable cases = 2C1*2C1 = 4 (1 freshman and 1 sophomore)
Probability = 4/24 = 1/6

Alternatively, we can arrange them in two spots since it doesn't matter in probability as long as we arrange both numerator and denominator.
Total number of cases = 8C1*6C1 = 48 (6C1 because the second person cannot be from the same grade)
Favorable cases = 4C1 * 2C1 = 8
Probability = 8/48 = 1/6

Answer (D)
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In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink] New post   12 Nov 2021, 08:06
2 freshmen, 2 sophomores, 2 juniors, and 2 seniors = Total eight students.

We need one freshmen and one sophomore. Favorable candidates = 2+2=4. Total candidates = 8.

S'pose, Probability of choosing one freshmen out of 8 possible candidates =\(\frac{ 4}{8}\). And probability of choosing one sophomore from 6 candidates (since 6 students left that we can choose from since we have got our freshman) = 2/6.

Combined Probability= \(\frac{4}{8} * \frac{2}{6} = \frac{1}{6}\) Option D.
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Re: In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink] New post   06 Jan 2022, 07:05
How many ways to pick the 2 classes to be represented

4!/2!2!= 6

Freshman/Sophomore class is 1 pairing

Probability 1/6

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Re: In a high school debating team consisting of 2 freshmen, 2 sophomores [#permalink]
06 Jan 2022, 07:05
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