Tanya bought 5 glasses for her kitchen - white, red, black, grey, and

She can select the glasses in 5C3 - 3C1 ways = 7 ways

Thank you for sharing your approach. The approach definitely looks good !

You divided the problem and used the principle of counting to obtain the total.

Here are few details on my working -

Given -
Tanya has white, red, black, grey, and blue.
She wants to arrange 3 of them on the shelf next to each other. Hence, she wants to arrange the glasses in a straight line.
Condition : A black and a white glass cannot be displayed at the same time.

Thus, there are two parts to this problem -
1) Selecting three glasses, out of the five that she bought (adhering to the condition)
2) Arranging the three selected glasses in a straight line.

Selecting three glasses

From a set of five glasses we can select three glasses in - 5C3 ways

However, this will also involve a selection where a black and a white glass have been selected. So we need to substract this from the selection.

Number of ways of selecting three glasses from a set of five glasses such that a black and a white glass is always selected = 3C1

How ?
After the black and the white glass is selected we have only selection to be done from the remaining 3 glasses.

So total ways of selecting the glasses = 5C3 - 3C1 ways

Arranging the three selected glasses

Three glasses need to be arranged in a straight line = 3! ways

Hence the total = (5C3 - 3C1) * 3! ways.

Hope this helps !