![]() ![]() 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. There are 3,326,400 ways to order the sheet of stickers.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. If we have a set of n objects and we want to choose r objects from the set in order, we write P\left(n,r\right). Before we learn the formula, let’s look at two common notations for permutations. Fortunately, we can solve these problems using a formula. The number of permutations of n distinct objects can always be found by n!.įinding the Number of Permutations of n Distinct Objects Using a Formulaįor some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Note that in part c, we found there were 9! ways for 9 people to line up. There are 362,880 possible permutations for the swimmers to line up. There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot. Draw lines for describing each place in the photo.Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first. There are 8 remaining options for second place, and then 7 remaining options for third place. We know Ariel must win first place, so there is only 1 option for first place. Multiply to find that there are 504 ways for the swimmers to place. Once first and second place have been won, there are 7 remaining options for third place. Once someone has won first place, there are 8 remaining options for second place. ![]()
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