A true false test has 15 questions. How many answer keys are possible of the answer to 6 of the questions is false and 9 are true?

I think this is a permutation and not a combination because each answer key would be different depending on the order of which are true or false.

I think the answer is 15P9 x 15P6

Am I right?

lets look at an example. A four q test, 3 t, 1 f

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4!/1! * 4!/3!=4*3*2*4 Nope..
What about 4!/3!1!

lets try that then with a five q, 3T, 2F
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5!/3!2!=5*2=10 And ten it is.

Thanks for typing that all out! It really helped.

Well, well, well... it seems like someone wants a lesson in clown math!

Actually, you're on the right track! Since the order matters (you're considering permutations), we can indeed use the formula for permutations to find the number of answer keys.

However, there's a tiny mistake in your formula. You said 15P9 x 15P6, but remember that when you have total of 15 questions and you're choosing 9 to be true, automatically the remaining 6 will be false. So, you don't need to calculate separate permutations for the true and false answers.

Therefore, the correct answer would be 15P9 (or 15P6, whichever you prefer), because once you choose which 9 questions are true, the remaining 6 answers will automatically be false.

Isn't math fun when you can clown around with it?

You are correct that this problem involves permutations because the order matters. However, your approach is not quite correct.

To determine the number of answer keys possible, we need to consider the positions of the true and false answers.

There are 15 positions, and we need to choose 6 of them for the false answers and the remaining 9 positions for the true answers.

The number of ways to choose 6 positions out of 15 for the false answers is given by 𝟭𝟱𝑪𝟲 (read as "15 choose 6"), which can be calculated as:

𝟭𝟱𝑪𝟲 = 15! / (6! * (15 - 6)!)

Similarly, the number of ways to choose 9 positions out of the remaining 9 positions for the true answers is given by:

𝟵𝑪𝟵 = 9! / (9! * (9 - 9)!)

Now, to obtain the total number of answer keys possible, we need to multiply these two values together:

𝟭𝟱𝑪𝟲 * 𝟵𝑪𝟵 = 𝟭𝟯,𝟵𝟮𝟱

Therefore, there are 13,925 answer keys possible if there are 6 false answers and 9 true answers.

No, you are not correct. The number of answer keys possible can be determined using combinations, not permutations.

In this case, you have 15 questions, and you want to select 6 of them to be false and the remaining 9 to be true. The order in which the true and false answers are arranged does not matter, so we are dealing with combinations.

The formula to calculate combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

where n is the total number of items and r is the number of items you want to choose.

In this case, you want to choose 6 out of 15 questions to be false, so the formula becomes:

C(15, 6) = 15! / (6! * (15 - 6)!)

Calculating this:

C(15, 6) = 5005

Therefore, there are 5005 possible answer keys when 6 questions are false and 9 are true.