## 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

## Introduction to Combinations

Introduction to Combinations

In my Introduction to Permutations post, we have learned that the number of permutations (or arrangements) of $n$ objects taken at $n$ at a time written as $P(n,n)$ is equal to $n! = n(n-1)(n-2) \cdots (3)(2)(1)$, and we have also learned that the number of permutations of $n$ objects taken $k$ at a time written as $P(n,k)$ is equal to $\displaystyle\frac{n!}{(n - k)!}$.

In Figure 1, shown are the permutations of $4$ letters, A, B, C and D taken $4$ at a time.  From the figure, we can see that there are indeed $4!= (4)(3)(2)(1) = 24$ of such arrangement. In Figure 2, shown are the permutations of $4$ letters taken $3$ at a time, and we have shown that the number of permutations is equal to $\displaystyle\frac{4!}{ (4 - 3)!} = \frac{4!}{1!} = (4)(3)(2)(1) = 24$.  In Figure 3, we have again listed the permutations of $4$ letters taken $2$ at a time, and have shown that the number of permutations is equal to $\displaystyle\frac{4!}{(4-2)!} =12$.

Figure 1 – Permutations of ABCD, taken 4 at a time.

Figure 2 – Permutations of ABCD, taken 3 at a time.

Figure 3 – Permutations of ABCD, taken 2 at a time.

If we talk about combinations, however, the arrangement of objects does not matter. For example, if we want to buy a milk shake and we are allowed to choose to combine any $3$ flavors from Apple, Banana, Cherry and Durian*, then the combination of Apple, Banana and Cherry is the same as the combination Cherry, Apple, Banana.

Try to list all the possible combinations of $3$ flavors taken from $4$ before proceeding.

If we choose to shorten the name the fruits by selecting the first letter of their names, we only have $4$ possible combinations for question above: ABC, ABD, ACD, and BCD. Notice that these are the only possible combinations. Also, observe that if we list the permutations of ABC, we have ACB, BAC,BCA, CAB and CBA.  This means that in permutations, we have counted each combination of $3$ flavors from $4$ flavors $6$ times (or $3!$ times instead of one.

In other words, a combination is just like a subgroup of a group. For instance, if we want to find the number of subgroups containing $3$ objects taken from $4$ objects (or the combination of $4$ objects taken $3$ at a time), it is the same as asking  “how many possible groups of $3$ objects can be taken from $4$ objects?” In Figure 4, all the possible subgroups of $3$ letters taken from $4$ letters are displayed by the orange border. You also would have realized that the number of permutations is an overcounting of the number of combinations.

Figure 4 – The combinations of 4 objects taken 3 at a time is the same as the number of subgroups of 3 objects taken from 4 objects.

In Figure 2, ABC, ACB, BAC, BCA, CAB and CBA are permutations of Apples, Banana and Cherry. For each subgroup of $3$, we realized that we counted $3! = 6$ times. So, to get the number of combinations, we divide our number of permutations $P(4,3)$ by the number of permutations of our subgroup$P(3,3) = 3!$. Therefore, we can say that the number of combinations of $4$ objects taken $3$ at a time is equal to

$\displaystyle\frac{P(4,3)}{P(3,3)} = \frac{\frac{4!}{(4-3)!}}{3!} = \frac {4!}{(4-3)! 3!}$

In general, to get the number of combinations of $n$ objects taken $k$ at a time, we have to divide the number of permutations of $P(n,k)$ by the number of permutations of the subgroup $P(k,k)$.

$\displaystyle\frac{P(n,k)}{P(k,k)} = \frac{\frac{n!}{(n-k)!}}{n!} = \frac {n!}{(n-k)! k!}$

The combinations of $n$ objects taken $k$ is usually denoted by $C(n,k)$ or $\displaystyle n \choose k$

_____________________________________________________________

*Durian is a fruit which can be found in the Philippines. It looks like a jackfruit.

1 2