Fill in the blank to write this product as a factorial.
[tex]$3 \cdot 2 \cdot 1=[?]!$[/tex]
Answer (2)
Calculate the product: 3 ⋅ 2 ⋅ 1 = 6 .
Recall the definition of a factorial: n ! = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ... ⋅ 2 ⋅ 1 .
Find the value of n such that n ! = 6 , which is n = 3 .
Therefore, 3 ⋅ 2 ⋅ 1 = 3 ! .
Explanation
Calculate the Product We are asked to express the product 3 ⋅ 2 ⋅ 1 as a factorial. Let's first calculate the product.
Compute the Product The product of 3 ⋅ 2 ⋅ 1 is: 3 ⋅ 2 ⋅ 1 = 6
Understanding Factorials Now, we need to find a number n such that n ! = 6 . Let's recall the definition of the factorial function: n ! = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ... ⋅ 2 ⋅ 1 . We can test a few small values of n to see if we can find the correct one.
Finding the Right Factorial If n = 1 , then 1 ! = 1 , which is not equal to 6. If n = 2 , then 2 ! = 2 ⋅ 1 = 2 , which is not equal to 6. If n = 3 , then 3 ! = 3 ⋅ 2 ⋅ 1 = 6 , which is equal to 6.
Final Answer Therefore, 3 ! = 3 ⋅ 2 ⋅ 1 = 6 . So, the missing value in the blank is 3.
Examples
Factorials are used in many areas of mathematics, including combinatorics, algebra, and calculus. For example, if you want to determine the number of ways to arrange 5 different books on a shelf, you would calculate 5! = 5 * 4 * 3 * 2 * 1 = 120. This means there are 120 different ways to arrange the books. Factorials also appear in probability calculations, such as determining the probability of winning a lottery or drawing specific cards from a deck.
The product 3 ⋅ 2 ⋅ 1 equals 6, which matches the value of 3 ! . Thus, we can express the product as 3 ! . Therefore, the answer is 3.
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