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About nPr Formula
- In the nPr formula, the letter "P" stands for "permutation," which signifies "arrangement."
- The nPr formula calculates the number of possible combinations for picking and arranging r items from a set of n items. Sometimes the layout is crucial.
- P (n, r) (or) n P r (or) n P r can be written as nPr. It's used to figure out how many distinct methods there are to choose and arrange r different things from n different things. The permutations formula is also known as the nPr formula (as we call a way of choosing and arranging things to be a permutation). Factorials are used in this formula. The nPr formula is shown below.
nPr Formula
The nPr formula is,
n! / (n - r)!=P (n, r) (or) nPr (or) n Pr
where
- n - total number of things
- r - number of things that have to be selected and arranged
nPr Formula Derivation
Consider n various things and the assumption that r separate objects should be chosen and ordered from them. Let's look into a few other options.
Because there are n things in total, there are n ways to choose the first one.
Because the first thing has already been picked, the number of options for selecting the second object is limited (n - 1).
Similarly, there are numerous options for selecting the third object (n - 2).
There are only (n - r + 1) objects remaining to choose from when choosing the rth object, hence it
- can be picked in (n - r + 1) ways.
- The number of ways (nPr) of picking and arranging r different objects from n different objects is, according to the fundamental counting principle.
- n (n - 1) (n - 2) ... (n - r + 1) = n P r
- We use factorial notations to make things easier. Multiplying and dividing (n - r) by the above expression... 3 • 2 • 1
- n P r = [n (n - 1) (n - 2) ...(n - r + 1) (n - r) ... 3 • 2 • 1] / [(n - r) ... 3 • 2 • 1] = n! / (n - r)!
- Thus, the nPr formula is derived.
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