Question

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?

$75$

$76$

$79$

$84$

$91$

Answer: B

Concepts:pigeonhole principle extremal argument

Difficulty rating: 1070

Solution:

Note that we can pull as many as $14$ balls of each color without ensuring that $15$ balls of one color are drawn.

This means that we can draw all of the black, white and blue balls, along with $14$ red, green, and yellow balls.

This gives us a total of $9 + 11 + 13 + 3 \cdot 14$ $= 33 + 42$ $= 75.$ We need to add one at the end, however, to ensure that we get that $15$ th ball of some color, $75 + 1 = 76.$

Thus, B is the correct answer.

2019 AMC 10A Problem 4

Answer: B

Difficulty rating: 1070

Solution:

Problem 4 in Other Years