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An equal number of juniors and seniors are trying out for

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An equal number of juniors and seniors are trying out for [#permalink] New post 05 Oct 2017, 20:38
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An equal number of juniors and seniors are trying out for six spots on the university debating team. If the team must consist of at least four seniors, then how many different possible debating teams can result if five juniors try out?

(A) 50
(B) 55
(C) 75
(D) 100
(E) 250

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[Reveal] Spoiler: OA
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Re: An equal number of juniors and seniors are trying out for [#permalink] New post 05 Oct 2017, 21:36
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Bunuel wrote:
An equal number of juniors and seniors are trying out for six spots on the university debating team. If the team must consist of at least four seniors, then how many different possible debating teams can result if five juniors try out?

(A) 50
(B) 55
(C) 75
(D) 100
(E) 250

Kudos for correct solution.


Here it is given as five juniors try out i.e we have 5 juniors and 5 seniors since equal number try out for the six spots

Now we can have two combination

first one - 4 seniors and 2 juniors = 5C4 * 5C2 =50 (We have to keep minimum nos of seniors to 4 as the statement says atleast 4 numbers of seniors are in the team)

Second one- 5 seniors and 1 juniors = 5C5 * 5C1 = 5

Therefore total possible teams = 50 + 5 =55
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Re: An equal number of juniors and seniors are trying out for [#permalink] New post 10 Jan 2019, 23:07
Expert's post
For anyone who gets stumped by which way to use combinatorics, lets think about this.

First,
They say that there are equal numbers of juniors and seniors, and there are 5 juniors, which means there are 5 seniors and a total of 10 players

Now, lets find the total number of all 6 team combinations out of 10 players, which is 10c6= 210, so it cant be greater than this number,
which means we eliminate E

Its said that there are at least 4 seniors that must be on the team, which means that there are 4 out of 5 seniors that have to be picked, leaving only 2 out of 5 Juniors that can be picked after.
picking 4 out of 5 seniors is 5c4,
picking 2 out of 5 juniors is 5c2
thus 5c4*5c2= 5*10= 50

but what if 5 out of 5 seniors get picked? then 5c5=1
leaving only 1 out of 5 juniors to be picked, 5c1=5
Thus 5c5*5c1= 1*5=5

add all the possible combinations to get 50=5=55

We choose B
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Re: An equal number of juniors and seniors are trying out for [#permalink] New post 01 Aug 2020, 06:19
Expert's post
Bunuel wrote:
An equal number of juniors and seniors are trying out for six spots on the university debating team. If the team must consist of at least four seniors, then how many different possible debating teams can result if five juniors try out?

(A) 50
(B) 55
(C) 75
(D) 100
(E) 250

Kudos for correct solution.


If the team must have AT LEAST 4 seniors, then we must consider two possible cases:
Case i: The team has 4 seniors and 2 juniors
Case ii: The team has 5 seniors and 1 junior

Case i: The team has 4 seniors and 2 juniors
Since the order in which we select the seniors does not matter, we can use combinations.
STAGE 1: We can select 4 seniors from 5 seniors in 5C4 ways (= 5 ways)
STAGE 2: We can select 2 juniors from 5 juniors in 5C2 ways (= 10 ways)
By the Fundamental Counting Principle (FCP), we can complete both stages in (5)(10) ways = 50 ways

Aside: See the video below to learn how to quickly calculate combinations (like 5C2) in your head

Case ii: The team has 5 seniors and 1 junior
STAGE 1: We can select 5 seniors from 5 seniors in 5C5 ways (= 1 way)
STAGE 2: We can select 1 junior from 5 juniors in 5C1 ways (= 5 ways)
By the Fundamental Counting Principle (FCP), we can complete both stages in (1)(5) ways = 5 ways

TOTAL number of outcomes = 50 + 5 = 55

Answer: B

Cheers,
Brent

VIDEO ON CALCULATING COMBINATION IN YOUR HEAD

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Brent Hanneson – Creator of greenlighttestprep.com
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Re: An equal number of juniors and seniors are trying out for   [#permalink] 01 Aug 2020, 06:19
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