Answer to Question #272591 in Discrete Mathematics for dv13

Question

8 A bag of chocolates has 20 milk chocolates, 20 dark chocolates and 20 white chocolates. 

a) What is the minimum number of chocolates you must select at random from the bag to guarantee that you will get at least 5 of the same kind of chocolates?

b) What is the minimum number of chocolates you must select at random from the bag to guarantee that you will get at least 5 milk chocolates?

9 Usually, a person doesn’t drink more than 3000 mL of water per day.  On a regular day, there are 10,500 people on some university campus.  At least how many people on campus would drink the exact same amount of water (in mL) on that day?

Answer 1

8.

a) The possible case is: 4 milk chocolates, 4 dark chocolates and 4 white chocolates. The next chocolate gives the necessary variant.

$$4+4+4+1=13$$

b) The possible case is: 20 dark chocolates and 20 white chocolates. The next 5 chocolates (milk) gives the necessary variant.

$$20+20+5=45$$

THE GENERALIZED PIGEONHOLE PRINCIPLE

If $N$ objects are placed into $k$ boxes, then there is at least one box containing at least $⌈N∕k⌉$ objects.

$$\dfrac{10500}{3000}=3+\dfrac{1500}{3000}$$

At least 4 people on campus would drink the exact same amount of water.