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.

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