What is the condition for ensuring that at least 14 jelly beans removed from a jar are of the same color?

To establish the condition for ensuring that at least 14 jelly beans removed from a jar are of the same color, it is essential to apply the principles of the pigeonhole theorem. This theorem suggests that if more items are distributed across a smaller number of categories than there are categories, at least one category must contain multiple items.

When considering option that involves removing n-1 beans of each color, if there are multiple colors of jelly beans (let's say k colors), by removing n-1 beans from each of those k colors, you would leave us one bean short of reaching the next full grouping of any color. Thus, when you remove one more jelly bean beyond this point—making it n (where n is the number of jelly beans of the color being targeted)—you are guaranteed to have at least one color reach 14 beans.

For example, if there are five colors of jelly beans, by removing 13 beans from each color, you have removed a total of 65 beans (assuming you started with at least 14 of each color) but have yet to exceed 13 in any individual color. However, with the next bean removed, you must pull one from a color that already has 13, thus securing that color