Decide whether each of the following statements is true. If false, demonstrate why. a. \( 6!=6 \cdot 5 \) ! b. \( 3!+3!=6! \) c. \( \frac{6!}{2!}=3 \) ! A. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The statement is true. B. The statement is false because \( 6!=\square \) and \( 6 \cdot 5!=\square \) (Simplify your answers.)

Let's evaluate each statement one by one and determine their truthfulness. a. \( 6! = 6 \cdot 5! \) This statement is true! By definition, factorials are calculated as the product of all positive integers up to that number. So, \( 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 6 \cdot 5! \). b. \( 3! + 3! = 6! \) This statement is false. To break it down, \( 3! = 3 \cdot 2 \cdot 1 = 6 \). Therefore, \( 3! + 3! = 6 + 6 = 12 \). However, \( 6! = 720 \). So, the correct fill-in would be: \( B. The statement is false because \( 3! + 3! = 12 \) and \( 6! = 720 \). c. \( \frac{6!}{2!} = 3! \) This statement is also false. First, \( 6! = 720 \) and \( 2! = 2 \). Therefore, \( \frac{6!}{2!} = \frac{720}{2} = 360 \). But \( 3! = 6 \). The correct fill-in would be: \( B. The statement is false because \( \frac{6!}{2!} = 360 \) and \( 3! = 6 \). So, here are the conclusions: a. True b. False (because \( 3! + 3! = 12 \) and \( 6! = 720 \)) c. False (because \( \frac{6!}{2!} = 360 \) and \( 3! = 6 \))