Calculate permutations without repetition

Permutation counting determines the number of ordered arrangements of $r$ distinct objects selected from $n$ available objects without repetition: $nP_r=n(n-1)\cdots(n-r+1)=\frac{n!}{(n-r)!}$ , where $0\le r\le n$ . The understanding distinguishes situations in which order creates different outcomes from combinations, connects the formula to the multiplication principle and factorial notation, and is limited to finite, nonrepeating arrangements rather than repeated-element, circular, or more abstract permutation forms.

Detailed Explanation: Calculate permutations without repetition

When you arrange $r$ distinct objects chosen from $n$ available objects without repetition, order matters. This is called a permutation.

Use the formula

nP_r=\frac{n!}{(n-r)!}

where $n!$ means factorial:

n!=n(n-1)(n-2)\cdots 1.

Example

A class has $8$ students. In how many ways can the teacher choose a president, a vice-president, and a secretary?

Because these are different positions, order matters. Choosing Ana as president and Ben as vice-president is different from choosing Ben as president and Ana as vice-president.

There are three positions to fill:

Choose the president: $8$ choices.
Choose the vice-president: $7$ remaining choices.
Choose the secretary: $6$ remaining choices.

By the multiplication principle,

8\cdot 7\cdot 6=336.

Using permutation notation gives the same result:

8P_3=\frac{8!}{(8-3)!} =\frac{8!}{5!} =\frac{8\cdot7\cdot6\cdot5!}{5!} =8\cdot7\cdot6 =336.

Therefore, there are

\boxed{336}

ways to assign the three positions.

Remember: use permutations when the objects are distinct, no object is used more than once, and changing the order creates a different outcome.

Learn by doing: Calculate permutations without repetition

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Probability Permutation or Combination - Scenario to Value

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