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
