Worst-case scenario.

oops... I made a mistake.

I meant:

Lucy was given a box containing 60 pens: 20 red, 20 blue, 15 white and 5 black.

At least how many pens must she take from the box in order to get 16 pens of one colour without looking into the box? Give out solutions.

Well she cannot get 16 black or white pens at all because there are not that many of them in the box. ://

I still do not think I understand what you are asking.

If she takes out 16 pens all of the same colour then that meets this requirement. So the answer is 16

I interpret this question as:

What is the minimum number of pens that must be selected (blindly) such that Lucy is guaranteed to have 16 pens of the same colour in the selection?

It is possible (though perhaps unlikely) that she selects in the following order:

5 black pens

15 white pens

15 blue pens

16 red pens (or possibly 15 red pens followed by 1 blue pen for these last 16)

The total number she has to select in order to guarantee getting 16 of one colour is therefore 51.

.

The minimum number of pens that she would have to draw to get 16 of one color would be 16  - she either selects 16 of the red ones on the first 16 draws or 16 of the blue ones on the first 16 draws......

The maximum number of pens that she would have to draw is given by this scenario :

She draws all of the black and white ones on the first 20 draws 

She then selects 15 of one color - either the red  or blue - on the next 15 draws......

Then......on the next 15 draws she selects the opposite color to the previous 15 draws .......this is  20 + 15 + 15   = 50 draws to this point

Finally.....on the next draw.....she must select either a red or a blue.......so.....51 draws....just as Alan said.....!!!!