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# help counting

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A teacher has made ten statements for a True-False test. Four statements are true and six are false. However, no two true statements are consecutive.  How many distinct answer keys could there be for the test?

Jan 7, 2021

#1
+303
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$$C^n_r=\frac{n!}{r!(n-r)!}$$

$$C^6_4=\frac{6!}{4!(6-2)!}$$

$$C^6_4=\frac{6!}{4!*2!}$$

$$C^6_4=\frac{6*5}{2!}$$

$$C^6_4=\frac{30}{2}$$

$$C^6_4=15$$

$$\boxed{15}$$

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Jan 7, 2021
#2
+112874
-1

A teacher has made ten statements for a True-False test. Four statements are true and six are false. However, no two true statements are consecutive.  How many distinct answer keys could there be for the test?

4 true

6 false

no 2 true in a row.

*T F* T F *T F* T  *     These  T and F ones are fixed and the stars represent where the other falses can go.

If all the other three falses are together then there are 5 places they can go.       5

If none of the other 3 falses are together then there is 5C3 = 10 places they can go.    10

If 2 of the falses are together and the other seperate then there are   5C2 *2 = 10*2 = 20 places they can go.

5+10+20 = 35 distinct answer keys

Jan 7, 2021