I had a hard time trying to convince myself of the derivation of the formula

nPr = P(n, r) = n!/(n-r)!

After reading and re-reading the read work, here we are:

Since a permutation involves selecting r distinct items without replacement from n items and order is important,

P (n, r) = n.(n-1).(n-2). (n-3) …. (n-r+1)——————————– (1)

Notice the numbers reducing from n until they reach the number (n-r+1). This last term (n-r+1) avoids a zero in case n=r.

Since (n-r)! / (n-r)! = 1, multiplying the right hand side of

Equation (1) by (n-r)! / (n-r)! results in:

P (n, r) = [n.(n-1).(n-2).(n-3)…. (n-r+1)] X (n-r)! / (n-r)!———– (2)

Since (n-r)! = (n-r).(n-r-1)….(3)(2)(1)————————–(3)

Equation (2) becomes:

P (n, r)=[n .(n-1).(n-2)…. (n-r+1)] X [(n-r).(n-r-1)….(3)(2)(1) / (n-r).(n-r-1)….(3)(2)(1)]————————————(4)

A closer look at the numerator of Equation (4) shows that the numbers are reducing from n to 1. Here is the numerator again:

[n .(n-1).(n-2)…. (n-r+1)] X [(n-r).(n-r-1)….(3)(2)(1)

n-1 is 1 less than n and (n-r) is one less than (n-r+1) and so on until 1 which is less than 2 by one.

We can safely conclude therefore that the numerator = n! Since n! =n. (n-1).(n-2)…..(3).(2).(1)

Equation (4)’s denominator (n-r).(n-r-1)….(3)(2)(1) is in effect Equation (3) which equals (n-r)!.

Hence the numerator is n! and the denominator is (n-r)!

Meaning:

nPr = P(n, r) = n!/(n-r)!

Proved

Books I read to arrive at this:

Statistics for Business and Economics- Paul Newbold

Statistics for Business and Economics-Frank Tailoka

great article… was looking for this!

There are some minor mistakes … In equations 1 and 2 you have missed (n-1)

Thank you for the feedback. (n-1) has been inserted into both equations.

THIS HELPED ME A LOT TO UNDERSTAND THE CONCEPT OF PERMUTATION AND THE DERIVATION OF ITS FORMULA

THANKS TO THE AUTHOR

You are welcome

Thanks a lot! You explained the equation 4 numerator so well! 🙂

Its easy to understand by separating like equations, thank you

This is superbly good, it has helped me so much in my assignment. Thank you so much.

Excellent step by step derivation. Thank you

Thanks for that. All the maths books I could find moved directly from nPr = n(n-1)(n-2)….(n-r+1) to nPr = n!/(n-r)! without explanation, and proving the identity was beyond me. All my books also explain how to get from the second permutation formula to nCr = n!/ (n-r)!r!.

I wonder why a full explanation of the derevation you elucidate so well is hardly ever given?

youre awesome!.. thanks a billion!