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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?

3

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Solution

The correct option is E $9$
The problem basically says, how many scarves must we pick to have a $100 %$ chance of there being $3$ or more of the same color. All we have to do to disprove an answer choice is give a scenario in which there are $2$ or less of the same color, based on the number of scarves we're choosing.
$3$ scarves
We could have $1$ red, $1$ brown, $14$ yellow scarf so A is wrong
$6$ scarves
We could have $2$ red, $2$ brown, $1$ yellow, $1$ gray so B is wrong
$7$ scarves
We could have $2$ red, $2$ brown, $2$ yellow, $1$ gray so C is wrong
$8$ scarves
We could have $2$ red, $2$ brown, $2$ yellow, $2$ gray so D is wrong
E is left as the correct answer. Even if we tried to minimize the number of repeating colors, at best we'd have $3$ of the same color and $2$ each of the rest $(3 + 2 + 2 + 2 = 9)$ .

Hence, option E is correct.

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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?

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?