What is the expression for calculating the factorial of a number n?

What is the expression for calculating the factorial of a number n?

• A. n(n-1)(n-2)...(2)(1)
• B. (n!)/(n-r)!
• C. (n!)/((n-r)!r!)
• D. n!

Answer: (A)

The factorial of a number \( n \), denoted as \( n! \), is defined as the product of all positive integers from \( n \) down to \( 1 \). This means that to calculate \( n! \), you multiply \( n \) by every integer that is less than \( n \) until you reach \( 1 \). The expression provided reflects this definition accurately. So, if you have a number, let’s say \( 5 \), the factorial \( 5! \) would be calculated as \( 5 \times 4 \times 3 \times 2 \times 1 \). This aligns perfectly with the described expression \( n(n-1)(n-2)...(2)(1) \), where you continue multiplying by each preceding integer until you reach 1. The other options given involve different mathematical concepts. For instance, options involving \( n! \) divided by any factorials relate to combinations or permutations, where factors represent selections or arrangements from a set, which do not apply to the factorial definition directly. Therefore, recognizing that \( n! \) specifically represents the product of the sequence back to 1 is key to understanding why the correct expression is the one

The factorial of a number ( n ), denoted as ( n! ), is defined as the product of all positive integers from ( n ) down to ( 1 ). This means that to calculate ( n! ), you multiply ( n ) by every integer that is less than ( n ) until you reach ( 1 ). The expression provided reflects this definition accurately.

So, if you have a number, let’s say ( 5 ), the factorial ( 5! ) would be calculated as ( 5 \times 4 \times 3 \times 2 \times 1 ). This aligns perfectly with the described expression ( n(n-1)(n-2)...(2)(1) ), where you continue multiplying by each preceding integer until you reach 1.

The other options given involve different mathematical concepts. For instance, options involving ( n! ) divided by any factorials relate to combinations or permutations, where factors represent selections or arrangements from a set, which do not apply to the factorial definition directly. Therefore, recognizing that ( n! ) specifically represents the product of the sequence back to 1 is key to understanding why the correct expression is the one