## Milkshakes and Power Sets

In Milkshakes, Beads, and Pascal’s Triangles, we have talked about  a systematic way of choosing a combination of objects from a larger number of objects. Let us recall the problem in the said post.

Issa went to a shake kiosk and want to buy a milkshake. The shake vendor told her that she can choose plain milk, or she can choose to combine any number of flavors in any way she wants. There are four flavors to choose from: Apple, Banana, Chico, and Durian. How many possible combination of flavors can Issa make?

In the problem, Issa can choose any number of flavors and any combination. She can choose plain milk, choose one flavor at a time, two flavors at a time, three flavors at a time, or four flavors at a time as shown in the table below (click the table to enlarge).

Notice that in writing the list, we have exhausted the number of subsets in a set with four elements.  If we let a, b, c, and d stand for avocado, banana, chico, and durian, let them be members or a set and use the set notation, we can write the subsets as follows: » Read more

## The Unfinished Game Problem

In the Milk, Beads and Pascal’s Triangle article, we have talked about the Sister’s Dilemma, and how they toss coins and later roll dice to solve their problem.  In this post, we are going to talk more about tossing coins and how they are connected to other mathematical topics.

The Interrupted Game

In a chess tournament in your school, two of your classmates, Sherwin and Carlo, made it to the championship game. The championship game was a race to 6. The score was 5-3, in favor of Sherwin.

Figure 1

Suppose, Carlo got sick, and the school agreed to divide the prize money worth \$500 based on the players’ chance of winning the championship, how should the money be divided fairly? » Read more

## The Binomial Expansion

Note: This is the second part of the Binomial Expansion Series

Part II: Binomial Expansion

In the Milkshakes, Beads, and Pascal’s Triangle article, we have shown that the combination of the binary numbers 1 and 0 may be interpreted as the number the flavors of milk shakes, or the number of possible paths of the bead in our Galton board as shown in Table 1. Recall that in the Milkshake problem, Issa was given a choice to combine any number of flavors from four fruits: Apple, Banana, Chico and Durian. Thus, 0101 means banana-durian milkshake. On the other hand, in the beads problem, 0101 is LRLR or the bead went to the left after hitting the peg in row A, right in after hitting the peg in row B, left after hitting the peg in row C and right after hitting the peg in row D. » Read more

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