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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