Bunuel wrote:
A lunar mission is made up of x astronauts and is formed from a total of 12 astronauts. A day before the launch the commander of the program decides to add p astronauts to the mission. If the total number of possible lunar missions remain unchanged after the commander’s decision, then which of the following cannot be the value of p?
(A) x
(B) x + 3
(C) 3
(D) 6
(E) 8
The total number of possible lunar missions before p astronauts are added is 12Cx. The total number of possible lunar missions after p astronauts are added is 12C(x+p). Since total number of possible lunar missions remain unchanged, we have:
12Cx = 12C(x+p)
Recall that we have a formula: nCx = nC(n-x). Since x + (n-x) = n and apply this to our equation, we have:
x + (x+p) = 12
2x + p = 12
Now let’s check the given answer choices (notice that we are looking for a value that can’t be p):
A) p = x
2x + x = 12
3x = 12
x = 4
This is not the choice we are looking for.
B) p = x + 3
2x + x + 3 = 12
3x = 9
x = 9
This is not the choice we are looking for.
C) p = 3
2x + 3 = 12
2x = 9
x = 4.5
Since x has to be an integer, then x can’t be 4.5, which means p can’t be 3.
Answer: C