Question

Note: Here you will demonstrate your ability to solve a basic combinations problem of the type you might encounter in a probability and statistics course.

a) Ben got up late one morning and was trying to find a clean pair of socks. He has a total of six pairs (three blue pairs and three white pairs, where each pair consists of two socks of the same color) in his drawer. However, Ben is not a very good housekeeper and his socks were just thrown in the drawer completely mixed up. The lightbulb in his room is burned out so he can't see, and it is still dark outside. Enumerate (i.e., list) all possible combinations of pulling out socks one by one from the drawer that will result in getting a matched pair of socks (i.e., either two blue socks or two white socks).
b) Suppose that Ben has nine pairs of socks (three blue pairs, three white pairs, and three purple pairs). List all of the possible combinations that produce a matched pair of socks (i.e., two blue socks, two white socks, or two purple socks).
c) Comment on the complexity of this problem as the number of available colors increases.

Hint: Drawing a picture that shows the possible sequences of socks being pulled from the drawer can be useful (think of a branching tree). If you need a more concrete example, use pieces of paper with the name of a color on it and pull them out from a box. What are all the possible ways you can do this to get a pair of matching socks?

Answer 1

Ben's possible combinations for finding a matched pair of socks involve pulling out similarly colored socks in sequence or trying different colors until he finds a match. As the number of sock colors increases, the complexity and the number of needed attempts also increase.

Step-by-step explanation:

When trying to solve a basic combinations problem where Ben is attempting to find a matched pair of socks in the dark, we consider the possible combinations of pulling out socks one by one from the drawer. Since each pair consists of two socks of the same color, there are certain steps Ben can take to ensure he gets a matched pair.

Enumerating Possible Combinations

In scenario (a), with three blue pairs and three white pairs, Ben's possible combinations to obtain a matched pair of socks by pulling out socks one by one are:

• Blue, Blue (BB)
• White, White (WW)
• Blue, White, Blue (BWB) or White, Blue, White (WBW)

In scenario (b), with an additional color, the possible combinations Ben can have to pull out a matched pair become:

• Blue, Blue (BB)
• White, White (WW)
• Purple, Purple (PP)
• Blue, White, Blue (BWB) or White, Blue, White (WBW) or Purple, Blue, Purple (PBP) or any other similar combination

As the number of colors increases, the complexity of the problem grows, making the pattern more intricate, and if we continue to add more colors, Ben will potentially have to pull more socks to ensure getting a matching pair.