In most likelihood, you will only get one or two combinations/permutations questions per test. Typically, the questions are quite challenging. Knowing the standard formulas won’t be enough; you’ll also have to use logic, and a fair amount of grit. Below is a question that will really force you to think, and rely on that grit. If you can get this question right in under 2 minutes, without a bead of sweat falling from your brow, then you are most likely ready for anything the GRE will throw you test day, combinations/permutations-wise.

Answer and explanation coming soon. Until then, let’s see who can get it. Good luck! ðŸ™‚

A space program is recruiting a team of astronauts to journey to Mars aboard a four-person shuttle. If the number of possible teams is less than 100 but greater than 10, then what is the possible number of astronauts who did not make the space team? IndicateÂ allÂ such numbers.

[A] Â 9

[B] Â 8

[C] Â 7

[D] 6

[E] Â 5

[F] Â 4

[G] Â 3

[H] 2

[I] Â Â 1

Answer and Explanation:Â

Two things to pay attention to here:

1)Â Â There will be four astronauts on the shuttle

2)Â Â We do not want the possible number the program is choosing from; we want the possible number of astronauts who do not make the team.

Letâ€™s call x the number we are choosing from. Using the combinations formula, since we are choosing for a team and the order does not matter, we get:

10 < x!/(x-4)!(4!) < 100

Finding the values for â€˜xâ€™, we get a range of 6 to 8. To illustrate:

6!/(6-4)!4! = 15

8!/(8-4)!4! = 70

We know x = 7 will fall inside the range. Therefore, x can equal 6, 7, or 8. Now hereâ€™s where we want to pay attention to 2). The number who will not make it will be x â€“ 4. Which yields, 2, 3, or 4 (Answers: F, G, H).

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I reached an answer of all except 6,7 and 8. I didn’t quiet catch the x-4 part. Can you please explain? Is it because the other values like 9 fall out of the range…?

Happy to help! To solve this question, we want to use the Combinations formula:

nCr = n!/((n-r)!r!)

which tells us the number of ways to choose r things from n things.

From the prompt, we know that we are choosing 4 astronauts for the shuttle, or in terms of the formula r = 4. Also, to solve the problem, Chris defines x as the number we are choosing from (n). When we plug in r=4 and n=x into the formula above we get:

xC4 = x!/((x-4)!4!)

And based on the restrictions of the problem, we can say that

10 < x!/(x-4)!(4!) < 100

When we solve for x, we get 6 < x < 8, where x is the number of astronauts we are choosing from. Since we're asked for the possible of number of astronauts who did not make the shuttle, we need to subtract 4, the number of astronauts chosen, to arrive at our final answer.

I chose 6, 7, 8 because I didn’t know what else to do after then. I didn’t think to apply the x-4 equation… :\ All that hard work to get right answers for the wrong question.

I’m not actually solving for ‘x’, in the traditional algebraic sense. I’m just providing a formula in which you should plug in a few different numbers to test the range of values. We want to find the smallest value for ‘x’ that yields a number larger than 10, and the largest value for ‘x’ that yields a number less than 100.

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Hi Chris,

I reached an answer of all except 6,7 and 8. I didn’t quiet catch the x-4 part. Can you please explain? Is it because the other values like 9 fall out of the range…?

Hi Sayahnita ðŸ™‚

Happy to help! To solve this question, we want to use the Combinations formula:

nCr = n!/((n-r)!r!)

which tells us the number of ways to choose r things from n things.

From the prompt, we know that we are choosing 4 astronauts for the shuttle, or in terms of the formula r = 4. Also, to solve the problem, Chris defines x as the number we are choosing from (n). When we plug in r=4 and n=x into the formula above we get:

xC4 = x!/((x-4)!4!)

And based on the restrictions of the problem, we can say that

10 < x!/(x-4)!(4!) < 100

When we solve for x, we get 6 < x < 8, where x is the number of astronauts we are choosing from. Since we're asked for the possible of number of astronauts who did not make the shuttle, we need to subtract 4, the number of astronauts chosen, to arrive at our final answer.

I hope this clears things up ðŸ™‚

And I thought I was sorted with Combinatorics!

I chose 6, 7, 8 because I didn’t know what else to do after then. I didn’t think to apply the x-4 equation… :\ All that hard work to get right answers for the wrong question.

Oh well, I have to keep optimistic!

Keep it up Kayden! You’ll be able to crack these super challenge questions soon :).

Can you explain how you solved for x, i formed the equation but couldn’t solve for x.

Hi Venkatesh,

I’m not actually solving for ‘x’, in the traditional algebraic sense. I’m just providing a formula in which you should plug in a few different numbers to test the range of values. We want to find the smallest value for ‘x’ that yields a number larger than 10, and the largest value for ‘x’ that yields a number less than 100.

Hope that helps!

GREAT Question CHRIS!

F,G and H or 2,3 and 4.

The answer is 1,2,3,4 and 5 ???

F, G & H ðŸ™‚

F, G , H answers??

team of 4, so team size nC4, no of 10< different teams <100, n= 6,7,8, so no of astronauts on earth 2 3 4

I guess the answers are 2, 3 and 4. Are they correct?

Ans is 2,3 and 4.

The answer is G,H,I

[B] and [F] ?

8,7,6 options B,C,D

they make the team, all others DON’T make the team and are the correct answers to this question asked

Ans: 2, 3 and 4?