Problem Solving

swerve 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 OA is C.

Source: Magoosh

Before the launch day:

The total number of possible lunar missions = 12Cx;

On the launch day:

The total number of possible lunar missions = (12 + p)Cx;

We are given that 12Cx = (12 + p)Cx

Note nCr = nC(n - r)

Thus, (12 + p)Cx = (12 + p)C(12 + p - x)

=> x = 12 + p - x

p = 2x - 12

Let's look at the options now.

A. x: Pluggin-in the value p = x in p = 2x - 12, we get x = 12, a valid value of x; thus, p can be x or 12

B. x + 3: Pluggin-in the value p = x + 3 in p = 2x - 12, we get x = 15, a valid value of x; thus, p can be x + 3 or 15

C. 3: Pluggin-in the value p = 3 in p = 2x - 12, we get x = 7.5, not a valid value of x since x is a positive integer; thus, p cannot be 3

Correct answer.

Though we got the answer, let's discuss option D and E.

D. 6: Pluggin-in the value p = 6 in p = 2x - 12, we get x = 9, a valid value of x; thus, p can be 6

E. 8: Pluggin-in the value p = 8 in p = 2x - 12, we get x = 10, a valid value of x; thus, p can be 8

The correct answer: C

Hope this helps!

-Jay
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swerve 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