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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?

Answer Choices

A. $3,360$
B. $3,500$
C. $3,762$
D. $3,780$
E. $3,920$

Steps

Explanation	Calculations	Help
$We first calculate the number of ways to choose one project manager from nine eligible candidates.$	$C (9, 1) = \frac{9 !}{1 ! 8 !} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 9$	Theory & Tactics Method Card PC3-C Click to view full details
$We then calculate the number of ways to choose two software engineers from seven eligible candidates, where the two positions are identical.$	$C (7, 2) = \frac{7 !}{2 ! 5 !} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = 21$	Theory & Tactics Method Card PC3-C Click to view full details
$Next, we calculate the number of ways to choose one designer from twenty eligible candidates.$	$C (20, 1) = \frac{20 !}{1 ! 19 !} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 20$	Theory & Tactics Method Card PC3-C Click to view full details
$Because the project manager selection, the software engineer selection, and the designer selection are all independent, we multiply these counts to find the total number of teams.$	$9 \times 21 \times 20 = 3, 780$	Theory & Tactics Method Card PC2-A Click to view full details

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Final Answer

D

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?

Answer Choices

A. $3,360$
B. $3,500$
C. $3,762$
D. $3,780$
E. $3,920$

Steps

Explanation	Calculations	Help
$We first calculate the number of ways to choose one project manager from nine eligible candidates.$	$C (9, 1) = \frac{9 !}{1 ! 8 !} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 9$	Theory & Tactics Method Card PC3-C Click to view full details
$We then calculate the number of ways to choose two software engineers from seven eligible candidates, where the two positions are identical.$	$C (7, 2) = \frac{7 !}{2 ! 5 !} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{( 2 \times 1 ) \times ( 5 \times 4 \times 3 \times 2 \times 1 )} = 21$	Theory & Tactics Method Card PC3-C Click to view full details
$Next, we calculate the number of ways to choose one designer from twenty eligible candidates.$	$C (20, 1) = \frac{20 !}{1 ! 19 !} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{1 \times ( 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 )} = 20$	Theory & Tactics Method Card PC3-C Click to view full details
$Because the project manager selection, the software engineer selection, and the designer selection are all independent, we multiply these counts to find the total number of teams.$	$9 \times 21 \times 20 = 3, 780$	Theory & Tactics Method Card PC2-A Click to view full details

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Final Answer

D