14. Sequences & Series
Review of Factorials
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Evaluate the expression.
A
0
B
1
C
1,820
D
43,680
Verified step by step guidance1
Step 1: Recognize that the given expression is a factorial division problem: \( \frac{16!}{12! \cdot 4!} \). Factorials are the product of all positive integers up to a given number.
Step 2: Simplify the numerator \( 16! \) by canceling out the \( 12! \) in the denominator. This leaves \( 16 \cdot 15 \cdot 14 \cdot 13 \) in the numerator.
Step 3: Write the remaining expression as \( \frac{16 \cdot 15 \cdot 14 \cdot 13}{4!} \). Recall that \( 4! = 4 \cdot 3 \cdot 2 \cdot 1 \).
Step 4: Simplify the denominator \( 4! \) to get 24. The expression now becomes \( \frac{16 \cdot 15 \cdot 14 \cdot 13}{24} \).
Step 5: Perform the division by simplifying the numerator and denominator step by step. Multiply the terms in the numerator and divide by 24 to find the final result.
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