# Factorial : n/(n-k) = n(n-1)(n-2) (n-k+1) - why?

• michonamona
In summary, the equation (A) n!/(n-k)! = n(n-1)(n-2)...(n-k+1) is true because it is an informal shorthand that is meant to stop at (n-k+1) and not include (n-2) as a factor. It can be derived by cancelling out terms in the expanded version of the equation and understanding that for small values of n and k, it is not possible to explicitly list all the terms.
michonamona
Why is the equation

(A) n!/(n-k)! = n(n-1)(n-2)...(n-k+1)

true?

For example, let n=4 and k=2, then

4!/2! = 4x3x2x1 / 2x1 = 4x3 = 12.

I understand this example, but I can't make the connection with this and the right-hand-side of equation (A).

For example, why is our example above not

4!/2! = 4(4-1)(4-2)...(4-2+1).

I know this doesn't make any mathematical sense, but I can't understand how the equation on the right-hand-side of (A) is derived.

M

The equation is an informal shorthand. You aren't supposed to include (n-2) as a factor in the case where n=4 and k=2. You are supposed to STOP at (n-k+1)=3.

Adding to what Dick wrote - it may become more obvious when you try to derive the equation.

$$\frac {n!} {(n-k)!} = \frac {n \times (n-1) \times (n-2) \times ... \times (n - k + 1) \times (n - k) \times (n - k -1) \times ... \times 3 \times 2 \times 1} {(n-k) \times (n-k-1) \times (n-k-2) \times ... \times 3 \times 2 \times 1}$$

Check what cancels out and what is left. And remember that when n and k are too small it is not possible to explicitly list all these terms.

## What is factorial and how is it calculated?

Factorial is a mathematical operation that calculates the product of all positive integers from 1 up to a given number. It is represented by the symbol "!". For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. It is calculated by multiplying the given number by each positive integer that comes before it.

## Why does the formula for factorial include (n-k) and (n-k+1)?

The formula for factorial, n! = n(n-1)(n-2)...(n-k+1), includes (n-k) and (n-k+1) because it represents the number of ways to choose k objects from a group of n objects, without repetition and order being important. These terms ensure that each object is only counted once and that the order in which they are chosen does not matter.

## What is the significance of the number k in the factorial formula?

The number k in the factorial formula represents the number of objects being chosen from a larger group. It is also known as the "sample size". The value of k is crucial in determining the number of permutations or combinations possible in a given scenario.

## Can the factorial formula be applied to negative numbers?

No, the factorial formula is only defined for positive integers. It cannot be applied to negative numbers or fractions.

## Are there any special cases or exceptions in the factorial formula?

Yes, the factorial formula has two special cases: 0! = 1 and 1! = 1. These values are defined to maintain the consistency of mathematical operations and to make certain equations and calculations simpler.

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