Question

Rafael is taking a test that contains a section of 12 true-false questions. How many of the possible groups of answers to these questions have at least 6 correct answers of true?
a.
2510
b.
2545
c.
2450
d.
2300


Please select the best answer from the choices provided

A
B
C
D

Answer 1

Answer:

A

Step-by-step explanation:

Rafael is taking a test that contains a section of 12 true-false questions. At least 6 correct answers of true means that 6 answers are true, 7 are true, 8 are true, 9 are true, 10 are true, 11 are true or 12 are true.

Then

$C_6^{12}=\dfrac{12!}{6!(12-6)!}=924,\\ \\C_7^{12}=\dfrac{12!}{7!(12-7)!}=792,\\ \\C_8^{12}=\dfrac{12!}{8!(12-8)!}=495,\\ \\C_9^{12}=\dfrac{12!}{9!(12-9)!}=220,\\ \\C_{10}^{12}=\dfrac{12!}{10!(12-10)!}=66,\\ \\C_{11}^{12}=\dfrac{12!}{11!(12-11)!}=12,\\ \\C_{12}^{12}=\dfrac{12!}{12!(12-12)!}=1.$

Hence there are

$924+792+495+220+66+12+1=2510$

possible groups.