The GMAT may test you on the arrangement of groups and sets, so you are likely to see some permutation and combination problems. When you calculate permutations, you figure out the number of ways the elements of a set can be arranged in specific orders. Determining combinations is similar to finding permutations, except that the order of the arrangements doesn’t matter.

Counting Methods

Principle of Multiplication

If an object is to be chosen from a set of m objects and a second object is to be chosen from a different set of n objects, then there are mn ways of choosing both objects simultaneously.

For example, each time a coin is flipped, there are two possible outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, then the experiment has 28 possible outcomes, where each of these outcomes is a list of heads and tails in some order.

Factorial

A factorial is the product of all natural numbers in the set of numbers from 1 through a particular number (n), which is the number of the factorial. A symbol that is used with the multiplication principle is the factorial. If n is an integer greater than 1, then n factorial, denoted by the symbol n!, is defined as the product of all the integers from 1 to n.

The factorial is useful for counting the number of ways that a set of objects can be ordered. If a set of n objects is to be ordered from 1st to nth, then there are n choices for the 1st object, (n-1) choices for the 2nd object, (n-2) choices for the 3rd object, and so on, until there is only 1 choice for the nth object.

For example, the number of ways of ordering the letters A, B, and C is 3!, or 6. These orderings are called the permutations of the letters A, B, and C.

The factorial of 0 is written as 0!, which always equals 1.

Permutation

Permutations problems ask you to determine how many arrangements of numbers are possible given a specific set of numbers and a particular order for the arrangements. Consider the elements of S = {a, b, c}. You can arrange these three elements in six different ways:

a b c, a c b, b a c, b c a, c a b, c b a

Even though each group contains the same elements, these groupings are completely different permutations because they convey different orderings of the three elements. Writing out the number of possible orderings of a set of three letters isn’t too difficult, but what if you had to figure out the number of orderings for a set of 11 numbers? Luckily, you can rely on factorials to figure out permutations.

The number of permutations of n objects is expressed as n!. Instead of writing the possible permutations for the set of three letters {a, b, c}, use a factorial. Three different elements (the letters in the set) arranged in as many different orders as possible look like this: 3!, which is equal to 3×2×1, which is equal to 6. So, the three elements have six permutations.

Permutations get a little more challenging when you have a fixed number of objects, n, to fill a limited number of places, k, and you care about the order the objects are arranged in.

A permutation is a selection process in which objects are selected one by one in a certain order. The number of permutations of n things taken k at a time is:

permutation

Combination

Permutation means how many different arrangements can be created from a group of items. Combination means how many different ways of selecting items from a larger group. You form a combination by extracting a certain number of persons or things from a larger total sample of persons and things. Unlike permutations, the order doesn’t matter with combinations, so combinations result in fewer possibilities than permutations.

A combination problem may ask you to find how many different teams, committees, or other types of groups can be formed from a set number of persons. The formula is the number of ways to choose k objects from a group of n objects when the order of the objects doesn’t matter:

combination

This formula is different from the one for permutation. Because you have a larger number in the denominator than with a permutation, the final number will be smaller.