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From a group of 21 astronauts that includes 12 people with

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From a group of 21 astronauts that includes 12 people with

by BTGmoderatorDC » Fri Apr 12, 2019 4:28 pm

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From a group of 21 astronauts that includes 12 people with previous experience in space flight, a 3-person crew is to be selected so that exactly one person in the crew has previous experience in space flight. How many different crews of this type are possible?

A. 432
B. 94
C. 864
D. 1330
E. 7980

OA A

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by ceilidh.erickson » Sat Apr 13, 2019 4:42 pm
To count the number of possible crews, we must calculate 2 things:
1. the number of ways to select 1 person with experience out of 12
2. the number of ways to select the remaining 2 people out of the remaining 9

Remember that when counting combinations in a diminishing pool in which order doesn't matter, the # of ways to select m items out of n total is:
$$\frac{n!}{\left(m!\right)\left(\left(n-m\right)!\right)}$$

1. the number of ways to select 1 person with experience out of 12:
$$\frac{12!}{\left(1!\right)\left(\left(11\right)!\right)}$$
... or simply 12. We really don't need to set it up that way when we're just picking one item!

2. the number of ways to select the remaining 2 people out of the remaining 9:
$$\frac{9!}{\left(2!\right)\left(\left(7\right)!\right)}$$
= 36

For each of the 12 persons chosen for #1, there are 36 possible groups of 2 non-experienced astronauts. So we multiply the two together:
(12)(36) = 432

The answer is A.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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