Permutations Examples on Order2

In this class, We discuss Permutations Examples on Order2.

The reader should have prior knowledge of permutation examples in order 1. Click Here.

Example:

Given characters a, b, c, d, e.

Permutations containing “a before b” and “a before c”.

The above statement can be written as a before b and c.

The condition has two possibilities: a, b, c or a, c, b.

Solution:

Total five positions.

Select three positions out of five in 5c3 ways.

The three positions are filled with a, b, c or a, c, b.

The remaining two positions are filled with 2! ways.

Total permutations are 2 * 5c3 * 2!

Example:

The characters are a, b, c, d, and e.

Permutations that contain “a before b” or “a before c”.

Solution:

Take the case a before b

We take two positions out of five and fill a and b.

The below diagram shows the selection a before b.

The possible ways are 5c2 * 3!

Here, a before b means a, c, d, b, e is one case.

The above case a before b is arranged. But a before c is also mentioned.

“a before c” is counted in the case a before c, and we are counting again.

In this type of situation, we take the complement method.

Take the situations b before a and c before a, which complement the conditions a before b or a before c.

Total permutations subtracted by complement permutations will give the actual required.

b before a and c before a is given as b, c, a or c, b, a.

Permutations = Total possibilities – Complement possibilities.

Permutations = 5! – 2 * 5c3 * 2!.