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Count systematically: list choices at each step and multiply. Use factorials only when items equal slots (10 boys, 10 chairs → 10!). With fewer slots, multiply top-down (10×9×8×7×6). For independent events, raise choices to events (578). Structure over formulas prevents mistakes and speeds solutions.
When faced with counting problems, many students commit basic mistakes or overcomplicate matters by trying to directly applying formulas rather than using logical reasoning. The key is to think step by step, to ask yourself how many choices are available at each stage, and then to multiply these choices together. This approach works whether you are arranging people in seats, selecting objects, or evaluating independent events. Take a simple example. If 10 boys are to be seated in a row of 10 chairs, the first boy has 10 choices, the next has 9, the third has 8, and so on until the last boy has only 1 choice. This gives us 10 factorial, or 10!, total arrangements. The same principle extends to more complex cases such as partial arrangements or questions involving multiple independent outcomes. For deeper learning, explore our GMAT preparation course and strengthen your skills with GMAT practice tests. Do remember to analyze your performance carefully and to learn from the questions that you get incorrect – beyond a point in prep, improvement mainly comes by analyzing your mistakes.
Counting questions test your ability to determine how many possible arrangements or selections exist in a situation. The guiding principle is to evaluate the number of choices available at each step and then multiply them together. This concept underlies factorials, permutations, and combinations, all of which frequently appear on the GMAT.
Suppose 10 boys must be seated in a row of 10 chairs.
The first boy has 10 options, the second has 9, the third has 8, and so on until the last boy has just 1 option.
Multiplying these values gives us 10 × 9 × 8 × … × 1 = 10!, which is the total number of possible seating arrangements.
This illustrates how factorial notation simplifies long multiplication.
Now imagine 10 boys but only 5 chairs.
Students often mistakenly jump to 10! or 5!, but neither applies here.
The correct method is to think from the perspective of the fewer quantity, which is the 5 chairs.
The first chair has 10 options, the second has 9, and so on until the fifth has 6.
Thus, the total arrangements equal 10 × 9 × 8 × 7 × 6 = 3,240.
In another variation, suppose there are 78 exam questions and each question can be answered in 5 ways.
Since every question is independent of the others, we multiply 5 for each of the 78 questions. The total number of ways is 5 × 5 × … (78 times) = 578.
This example shows how exponents can express repeated multiplication efficiently.
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When you look closely, counting arrangements is not just a mathematical exercise but a metaphor for life’s choices. Each step depends on the options you had before, and the total outcome emerges from the discipline of multiplying these thoughtful decisions together. GMAT preparation mirrors this: you progress by breaking problems into manageable stages, analyzing patterns, and building clarity rather than memorizing formulas. The B-school admissions journey is similar, where structured thinking helps you handle complexity with confidence. In life too, progress comes not from shortcuts but from systematic, stepwise effort that ultimately shapes a meaningful and well-arranged path forward.
