Fill in the blank to write this product as a factorial.
[tex]$7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=[?]!$[/tex]
Answer (2)
Recognize that the given product is the product of all positive integers from 1 to 7.
Recall the definition of a factorial: n ! = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ... ⋅ 2 ⋅ 1 .
Express the product as a factorial: 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 7 ! .
The answer is 7 .
Explanation
Understanding the Problem We are given the product 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 and asked to express it as a factorial.
Definition of Factorial Recall that a factorial, denoted by n ! , is the product of all positive integers less than or equal to n . That is, n ! = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ... ⋅ 2 ⋅ 1 .
Identifying the Factorial In our case, we have the product of all positive integers from 1 to 7. Therefore, this product is equal to 7 ! .
Final Answer Thus, 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 7 !
Examples
Factorials are used in many areas of mathematics, including combinatorics, algebra, and calculus. For example, if you have 7 different books and you want to arrange them on a shelf, the number of different arrangements is 7! = 5040. This is because you have 7 choices for the first book, 6 choices for the second book, 5 choices for the third book, and so on, until you have only 1 choice for the last book. The total number of arrangements is the product of these choices, which is 7!. Factorials also appear in probability calculations, such as calculating the probability of winning a lottery or drawing specific cards from a deck.
The product 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 is equal to 7 ! . This is because the definition of a factorial includes multiplying all positive integers up to a certain number, which in this case is 7. So, the answer is 7 .
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