Question: Evaluate the expression. If the answer is not an integer, round to four decimal places. 8C4 …

Evaluate the expression. If the answer is not an integer, round to four decimal places. 8C4 … Select the correct factorial formula below and, if necessary, fill in the answer boxes to complete your choice. A. The factorial formula for nCr = n! (n - r)! , therefore 8C4 = (□ - □)! B. The factorial formula for nCr = n! r!(n - r)!, therefore 8C4 = (□ - □)! C. The factorial formula for nCr = n! r! , therefore 8C4 = □!

Solution

The problem is to evaluate \( \binom{8}{4} \) using the correct factorial formula. The formula for combinations is: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] We will use this formula to compute \( \binom{8}{4} \). First, identify the values: \( n = 8 \), \( r = 4 \). Now, apply the formula: \[ \binom{8}{4} = \frac{8!}{4!(8-4)!} \] \[ = \frac{8!}{4! \times 4!} \] Calculate the factorials: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] Substitute these values back into the formula: \[ \binom{8}{4} = \frac{40320}{24 \times 24} \] \[ = \frac{40320}{576} \] \[ = 70 \] Thus, the value of \( \binom{8}{4} \) is 70. This corresponds with option A.