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

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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.
  • #1
michonamona
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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.

Thanks for your help.

M
 
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  • #2
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.
 
  • #3
Adding to what Dick wrote - it may become more obvious when you try to derive the equation.

[tex]\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} [/tex]

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.
 

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

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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