there are 8 people participating in a focus group for a new software product related to the health system, 3 of them are software engineers, 2 of them are nurses, 1 of them is a doctor, and the remaining 2 people are technicians. in how many ways they can be seated in a row so that no two software engineers are together?

An alternative name for a combo is a choice. A combination is a choice made from a predetermined group of options. I won't organize anything here. They will be my choice. The number of distinct r selections or combinations from a set of n objects is indicated by the symbol [tex]^nC_{r}[/tex].

There are eight participants in the focus group, including three software engineers, two nurses, one doctor, and two technicians.

So the 3 software engineers =3! ways, 2 nurses =2! ways,

doctor =1! way , 2 technicians =2! ways.

We must determine how many different configurations are possible so that no two pieces of software may coexist.

In order to prevent two software engineers from being seated next to each other, we first arrange five persons in a row with a space between them.

[tex]We get that in 6 places they can sit in ^6C_{3} ways\\ xi.e ^6C_3 = 20 (by formula of combination)\\ Therefore total ways are,6 X 2 X 1 X2 X 20 = 480.[/tex]

Therefore, there are 480 different ways to arrange individuals in a row so that no two engineers sit together.

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