Evaluate the expression.
$\frac{16!}{12!\cdot4!}$

• A. 0
• B. 1
• C. 1,820
• D. 43,680

Answer: (C)

Verified step by step guidance

1. Understand that the expression \( \frac{16!}{12! \cdot 4!} \) is a combination formula, specifically \( \binom{16}{4} \), which represents the number of ways to choose 4 items from 16 without regard to order.
2. Recall the formula for combinations: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). In this case, \( n = 16 \) and \( r = 4 \).
3. Substitute the values into the combination formula: \( \binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!} \).
4. Simplify the factorial expression by canceling out the common terms in the numerator and the denominator. This involves recognizing that \( 16! = 16 \times 15 \times 14 \times 13 \times 12! \), allowing \( 12! \) to cancel out.
5. Calculate the remaining product in the numerator: \( 16 \times 15 \times 14 \times 13 \) and divide by \( 4! = 4 \times 3 \times 2 \times 1 \) to find the number of combinations.