In how many ways can you choose your entree? 10 ways Solution: In how many ways can you choose an appetizer? 6 ways In how many ways can one make a meal if one chooses one appetizer, one entree and one dessert? Let’s look at some more examples of basic counting principle.Įxample 2: A restaurant serves 6 varieties of appetizers, 10 different entrees and 4 different desserts. The distinction between ‘AND’ and ‘OR’ is quite important since it defines whether you will multiply or add. just one person), we could have done it in 3+2 = 5 ways because there are 5 people and we have to choose one of them.
If we had to choose one boy OR one girl (i.e. For every boy, we could choose a girl in 2 ways and there are 3 boys so we can choose a pair in 3*2 ways. This is so because we have to choose a boy AND a girl simultaneously. The word ‘distinct’ is important here as we will see in the next few weeks.Īlso notice here that it is not 3+2 = 5 ways. These spots can be filled in 3*2 = 6 ways. There are 3 contenders for the empty ‘boy spot’ and 2 contenders for the empty ‘girl spot’. Here we have 1 spot for a boy and 1 spot for a girl i.e. The basic counting principle deals with problems having ‘distinct spots’ and ‘available contenders’. We see that we can select a boy in 3 ways (since there are 3 boys) and we can select a girl in 2 ways (since there are 2 girls). Say the 2 girls are G1 and G2.Ī total of 6 ways. This is a very basic and very important concept. Solution: Let’s discuss the solution in detail. We want to select a pair of one boy and one girl for a dance. Let’s try to understand it using an example.Įxample 1: There are 3 boys and 2 girls. Also, many of the 700+ level questions use basic counting principle as the starting point so it’s not possible to start a discussion on combinatorics without discussing this principle first. The first thing I want to discuss is something we call “Basic Counting Principle” because it is useful in almost all 600-700 level questions of Combinatorics (Note here that I will avoid using the terms “Permutation” and “Combination” and the formulas associated with them since they are not necessary and make people uncomfortable). Remember to ask yourself whether order matters in the problem, and don’t forget the Fundamental Counting Principle! The GMAT may also combine one or more of these concepts in a longer Word Problem to make the question more challenging, but if you can remember these basics, you’ll be good to go! Since the order in which the athletes finish matters, we know to use the Permutation formula: Harder permutations problems will require you to use this formula:įor example, how many possible options are there for the gold, silver, and bronze medals out of 12 athletes? Here n = 12 and r = 3.
Therefore, there are 4 x 3 x 2 x 1 = 24 ways. For the next spot we’ll have 3, for the third spot we’ll have 2, and the last remaining person will take the final spot. How many different ways can four people sit on a bench? For the first spot on the bench, we have 4 to choose from. Therefore, the total number of possibilities is 9 x 2 x 1 = 18. The units digit has only 1 possibility (1). The tens digit has only 2 options (6 or 9). The hundreds digit can be any of these except 0 (since a three-digit number cannot begin with 0). Each digit has 10 possible values (0 through 9). To solve, we need to find the possible outcomes for each digit (hundreds, tens, and units) and multiply them. For example, how many three-digit integers have either 6 or 9 in the tens digit and 1 in the units digit? The Fundamental Counting Principle states that if an event has x possible outcomes and a different independent event has y possible outcomes, then there are xy possible ways the two events could occur together. This advanced concept is not as commonly tested as algebra fundamentals or number properties, but it’s definitely worth knowing the basics in case you do see it. Aiming for a 700+ on the GMAT? You never know when a challenging combination or permutation question will pop up three-quarters of the way through your exam to wreck havoc on your score.
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