SOLUTION: There are 10 true-false questions and 20 multiple choice questions from which to choose a ten question quiz. How many ways can the quiz be selected if there must be six true-false

Question 355687: There are 10 true-false questions and 20 multiple choice questions from which to choose a ten question quiz. How many ways can the quiz be selected if there must be six true-false questions and four multiple choice questions?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the number of ways you can select 4 multiple choice questions out of 20 multiple choice questions is:

20! / (4!*16!) = 4845

the number of ways you can select 6 true false questions out of 10 true false questions is:

10! / (6! * 4!) = 210

The quiz can be selected 210 * 4845 = 1017450 ways

to see how this works, use much smaller number.

assume you choose 1 TF out of 3 TF (TF = true false)
asume you choose 2 MC out of 4 MC (MC = multiple choice

The number of ways you can choose 1 TF out of 3 TF is:

3! / (1! * 2!) = 3

If you let a,b,c be each of the 3 TF choices, then those ways are:

a
b
c

The number of ways you can choose 2MC out of 4 MC is:

4! / (2! * 2!) = 6

if you let 1,2,3,4 be each of the 4 MC choices, then those ways are:
1,2
1,3
1,4
2,3
2,4
3,4

Multiply 3 * 6 and you get 18 total ways.

Those ways are:

a,1,2
a,1,3
a,1,4
a,2,3
a,2,4
a,3,4
b,1,2
b,1,3
b,1,4
b,2,3
b,2,4
b,3,4
c,1,2
c,1,3
c,1,4
c,2,3
c,2,4
c,3,4

The same concept applies, just with larger numbers.

You are using the combination formula.

It is:

C[n] = c! / (x! * (n-x)!)

n is the total set of possibilities.
x is the set of possibilities you are choosing out of the total set of possibilities.

With your Multiple Choice questions:
n = 20
x = 4
(n-x) = 16

With your true false questions:

n = 10
x = 6
(n-x) = 4

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