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?
+285 \(C^n_r=\frac{n!}{r!(n-r)!}\)
\(C^6_4=\frac{6!}{4!(6-2)!}\)
\(C^6_4=\frac{6!}{4!*2!}\)
\(C^6_4=\frac{6*5}{2!}\)
\(C^6_4=\frac{30}{2}\)
\(C^6_4=15\)
\(\boxed{15} \)
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+118608 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
