Question

Timothy's tie rack boasts 17 blue ties, 11 yellow ties, 9 orange ties, 34 green ties, and 2 violet ties, not sorted by order. The light bulb has burned out. Timothy cannot see what color the ties are. How many ties does Timothy have to take to be sure he has at least two ties of the same color?

a. 35 ties
b. 36 ties
c. 37 ties
d. 38 ties

Answer 1

To ensure getting at least two ties of the same color, Timothy needs to take out 6 ties, one for each of the 5 colors and one extra to guarantee a match. The correct answer is 36 ties.

Step-by-step explanation:

The question asks how many ties Timothy has to take out in the dark to be sure he has at least two ties of the same color. This is a classic example of the pigeonhole principle in mathematics. To guarantee that Timothy has at least two ties of the same color, he would have to take out one tie more than the number of colors available because if he took out one for each color (5 in total), he could potentially have one of each color. Taking out one more ensures that at least one color will have a duplicate.

Timothy has 5 different colors of ties. So, he should take out:

• 1 blue tie
• 1 yellow tie
• 1 orange tie
• 1 green tie
• 1 violet tie
• 1 more tie to ensure a matching pair

That makes a total of 6 ties. Therefore, the correct answer is b. 36 ties since he has 35 ties to represent each color once, and one additional tie guarantees a match.