Express $_{n}P_{n-4}$ without using factorials. Assume n > 4. Choose the correct answer below. A. $_{n}P_{n-4} = n(n-1)(n-2)(n-3)$ B. $_{n}P_{n-4} = n(n-4) \cdot \cdot \cdot 3 \cdot 2 \cdot 1$ C. $_{n}P_{n-4} = n(n-1)(n-2) \cdot \cdot \cdot 5$ D. $_{n}P_{n-4} = n(n-1)(n-2)(3)(n-4)$

In mathematics, particularly in permutations, the expression n ​ P r ​ represents the number of ways to arrange r items out of n items in a specific order, without repetitions. The formula for permutations is given by:
n ​ P r ​ = ( n − r )! n ! ​
However, the problem here is to express n ​ P n − 4 ​ without using factorials and select the correct expression from the given options. Let's break this down step-by-step:

Understand n ​ P n − 4 ​ :


Here, r = n − 4 , meaning we want to find the permutations of ( n − 4 ) elements from n total elements.


Substitute into the permutation formula:


n ​ P n − 4 ​ = ( n − ( n − 4 ))! n ! ​ = 4 ! n ! ​


Simplifying the expression without factorials:


Since 4 ! n ! ​ implies removing the last four terms from the factorial expansion of n ! , you need to calculate the product of the first n terms down to ( n − 4 + 1 ) terms, which is:

n × ( n − 1 ) × ( n − 2 ) × ( n − 3 )



Choose the correct multiple-choice answer:


Among the options provided, A : n ​ P n − 4 ​ = n ( n − 1 ) ( n − 2 ) ( n − 3 ) accurately reflects this simplified form.

Therefore, the correct option is A.

Answered by OliviaLunaGracy | 2025-07-21