A technology startup will select 1 project manager from 9 candidates, 2...

Question

A technology startup will select 1 project manager from 9 candidates, 2 software engineers from 7 candidates, and 1 designer from 20 candidates to form a core team of 4 professionals. If no candidate is eligible for more than one role, how many different teams are possible?

GMAT Panda · Accepted Answer

D We first calculate the number of ways to choose one project manager from nine eligible candidates. C(9,1) = 9!/1!8! = 987654321/1(87654321) = 9 We then calculate the number of ways to choose two software engineers from seven eligible candidates, where the two positions are identical. C(7,2) = 7!/2!5! = 7654321/(21)(54321) = 7654321/(21)(54321) = 21 Next, we calculate the number of ways to choose one designer from twenty eligible candidates. C(20,1) = 20!/1!19! = 2019181716151413121110987654321/1(19181716151413121110987654321) = 20 Because the project manager selection, the software engineer selection, and the designer selection are all independent, we multiply these counts to find the total number of teams. 9 21 20 = 3,780

Explanation	Calculations	Help
$We first calculate the number of ways to choose one project manager from nine eligible candidates.$	$C (9, 1) = \frac{9 !}{1 ! 8 !} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 9$
$We then calculate the number of ways to choose two software engineers from seven eligible candidates, where the two positions are identical.$	$C (7, 2) = \frac{7 !}{2 ! 5 !} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = 21$
$Next, we calculate the number of ways to choose one designer from twenty eligible candidates.$	$C (20, 1) = \frac{20 !}{1 ! 19 !} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 20$
$Because the project manager selection, the software engineer selection, and the designer selection are all independent, we multiply these counts to find the total number of teams.$	$9 \times 21 \times 20 = 3, 780$

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