Question

Marie is taking a test that contains a section of 10 true-false questions. How many of the possible groups of answers to these questions have at least 5 correct answers of true? Hint: Assign the variable x in the binomial expansion to be the number of true answers and y to be the number of false answers.

Answer 1

To solve this problem, we use the combination equation to find for the possible groups of answer to the questions. Since we are looking for at least 5 correct answers out of 10 questions, therefore we find for 10 ≥ r ≥ 5. We use the formula for combination:

nCr = n! / r! (n – r)!

Where,

n = total number of questions = 10

r = questions with correct answers

For 10 ≥ r ≥ 5:

10C5 = 10! / 5! (10 – 5)! = 252

10C6 = 10! / 6! (10 – 6)! = 210

10C7 = 10! / 7! (10 – 7)! = 120

10C8 = 10! / 8! (10 – 8)! = 45

10C9 = 10! / 9! (10 – 9)! = 10

10C10 = 10! / 10! (10 – 10)! = 1

Summing up all combinations will give the total possibilities:

Total possibilities = 252 + 210 + 120 + 45 + 10 + 1 = 638

Answer: 638