Use the formula for nCr to evaluate the expression. 9) 7C4

Answer = The combination formula, denoted as \( \binom{n}{r} \) or \( nCr \), is used to calculate the number of ways to choose \( r \) items from \( n \) items without regard to the order of selection. The formula for \( \binom{n}{r} \) is given by: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\] In this problem, we need to evaluate \( 7C4 \). Plugging \( n = 7 \) and \( r = 4 \) into the formula, we get: \[\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!}\] Now, calculate the factorials: \[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\] \[4! = 4 \times 3 \times 2 \times 1\] \[3! = 3 \times 2 \times 1\] Substituting these values into the formula, we get: \[\binom{7}{4} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1) \times (3 \times 2 \times 1)}\] Simplify by canceling the common terms in the numerator and the denominator: \[\binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1}\] Calculate the numerator and the denominator: \[7 \times 6 \times 5 = 210\] \[3 \times 2 \times 1 = 6\] Thus, \[\binom{7}{4} = \frac{210}{6} = 35\] Therefore, \( 7C4 = 35 \).