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Departments A, B, and C have 10 employees each, and department D has

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Departments A, B, and C have 10 employees each, and department D has [#permalink]

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New post 16 Jul 2016, 15:01
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Departments A, B, and C have 10 employees each, and department D has 20 employees. Departments A, B, C, and D have no employees in common. A task force is to be formed by selecting 1 employee from each of departments A, B, and C and 2 employees from department D. How many different task forces are possible?

A. 19,000
B. 40,000
C. 100,000
D. 190,000
E. 400,000
Spoiler: OA

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Re: Departments A, B, and C have 10 employees each, and department D has [#permalink]

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New post 16 Jul 2016, 15:41
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\(\begin{align}&\text{Department A, B, C}&&10\text{ Employees}&&\text{select }1\\
&\text{Department D}&&20\text{ Employees}&&\text{select }2
\end{align}\)

\(\text{Total }= {10 \choose 1}^3 \times {20 \choose 2} =\\
10^3 \times \frac{20!}{18!\times 2!} =\\
1,000 \times \frac{20 \times 19 \times 18!}{18! \times 2 \times 1} =\\ 1,000 \times \frac{20 \times 19}{2} =\\ 1,000 \times 190 = 190,000\\
\text{Answer: (D) 190,000}\)
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Re: Departments A, B, and C have 10 employees each, and department D has [#permalink]

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New post 16 Jul 2016, 15:49
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Bunuel wrote:
Departments A, B, and C have 10 employees each, and department D has 20 employees. Departments A, B, C, and D have no employees in common. A task force is to be formed by selecting 1 employee from each of departments A, B, and C and 2 employees from department D. How many different task forces are possible?

A. 19,000
B. 40,000
C. 100,000
D. 190,000
E. 400,000


Take the task of creating the task force and break it into stages.

Stage 1: Select one person from department A
There are 10 people to choose from, so we can complete stage 1 in 10 ways

Stage 2: Select one person from department B
There are 10 people to choose from, so we can complete stage 2 in 10 ways

Stage 3: Select one person from department C
There are 10 people to choose from, so we can complete stage 3 in 10 ways

Stage 4: Select 2 people from department D
Since the order in which we select the 2 people does not matter, we can use combinations.
We can select 2 people from 20 people in 20C2 ways (190 ways)
So, we can complete stage 4 in 190 ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create a task force) in (10)(10)(10)(190) ways (= 190,000 ways)

Answer:
Spoiler: ::
D


If anyone is interested, here's a video on calculating combinations (like 20C2) in your head

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Re: Departments A, B, and C have 10 employees each, and department D has [#permalink]

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New post 17 Jul 2016, 02:24
10c1 * 10c1* 10c1*20c2 = 10*10*10*190=190,000. Hence D is the correct answer.
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Re: Departments A, B, and C have 10 employees each, and department D has [#permalink]

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New post 06 Aug 2017, 20:40
Number of ways to choose from Department A,B and C : 10c1 = 10 each

Number of ways to choose from Department D(2 of 20) : 20c2 = 20*19/2 = 190

Total number of task forces possible are : 190*10*10*10 = 190000(Option D)
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Re: Departments A, B, and C have 10 employees each, and department D has   [#permalink] 06 Aug 2017, 20:40
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