Question

There are 6 red, 6 brown, 6 yellow, and 6 gray scarves packaged in 24 identical, unmarked boxes, 1 scarf per box. What is the least number of boxes that must be selected in order to be sure that among the boxes selected 3 or more contain scarves of the same color?

Answer 1

Hint: Here, we will solve these types of questions by picking one item each from the box and continue doing so till we get the scenario where the given condition is satisfied.

Complete step-by-step solution:
Given that there are 6 red, 6 brown, 6 yellow, and 6 gray scarves, which are packaged in 24 identical and unmarked boxes, 1 scarf per box.

Let us assume that the R represents the red color, B represents the brown color, Y represents the yellow color and G represents the gray color.

We will now solve by assuming that we are trying to get 3 of the same color as quickly as possible by imagining the absolute worst-case scenario.

First, we have picked one scarf of each color after 4 picks, that is, 1 of red color scarf R, 1 of brown color scarf B, 1 of yellow color Y and 1 of gray color G.

After these four picks, we have found out that this is going very inefficiently.

Now we have picked one scarf again from each and even after 8 picks, we have two scarfs each, that is, 2 of red color scarf, 2 of brown color scarf, 2 of yellow color scarf and 2 of gray color scarf.

Since, even now we don’t have 3 one color scarfs, so this is also not correct.

But if we will pick for one more time, this will definitely give us a 3 of one color scarf.

So the least number that must be selected to be 100 percent certain that you have 3 or more of the same color is 9.

Since it is possible to make 8 picks and still not have 3 of the same color. It is not possible to do so with 9 picks.

Note: In solving these types of questions, you should be familiar with the permutations. Students should also remember the properties for ease. Then use the given conditions and values given in the question, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.