GMAT Probability and Combinations: Complete Guide

Published on 2025-06-16 • 10 min read

Quick Takeaways

The Big Q: Does order matter? Yes = Permutation. No = Combination.
Reflex: 'At least one' = 1 - P(None). (Memorize this!).
Slot Method: Often simpler than P/C formulas for arrangements.
Rules: AND means Multiply; OR means Add (if mutually exclusive).
Glue Method: Keep items together by treating them as one unit.

Combinatorics & Probability: Counting and Chance

Probability and Combinatorics are two sides of the same coin. Combinatorics (which includes combinations and permutations) is the art of 'smart counting'—figuring out the total number of possible outcomes. Probability then takes it a step further, asking for the likelihood of a specific outcome. On the GMAT, these topics are often combined, requiring you to first count the possibilities and then determine the probability.

Combinations vs. Permutations: Does Order Matter?

This is the single most important question you must ask yourself in any GMAT counting problem. The answer determines which method you use.

Permutations: When Order Matters

Use permutations when the arrangement or sequence of items is important. Think about arranging books on a shelf, assigning people to specific roles (like President and VP), or creating a code. A-B is different from B-A. The Slot Method is often the most intuitive way to solve permutation problems: draw a blank for each 'slot' you need to fill, write the number of options for each slot, and multiply.

Combinations: When Order Doesn't Matter

Use combinations when you are simply choosing a group of items and the order of selection is irrelevant. Think about picking a team, choosing toppings for a pizza, or selecting a committee. A team of Alice and Bob is the same as a team of Bob and Alice. The formula for combinations is: nCr = n! / [r!(n-r)!].

The Rules of Probability

All probability questions on the GMAT boil down to one simple formula: Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes).

The 'AND' Rule (Independent Events): The probability of two independent events both happening is the product of their individual probabilities: P(A and B) = P(A) × P(B). For example, the probability of flipping heads twice is 1/2 × 1/2 = 1/4.
The 'OR' Rule (Mutually Exclusive Events): The probability of either of two mutually exclusive events happening is the sum of their individual probabilities: P(A or B) = P(A) + P(B).
The Complement Rule (The 'At Least One' Trick): The probability of an event happening is 1 minus the probability of it not happening: P(A) = 1 - P(Not A). This is a powerful shortcut for 'at least one' problems. For example, the probability of getting at least one head in three coin flips is 1 - (the probability of getting zero heads).

Advanced Strategies for Tough Problems

The 'Glue' Method for 'Together' Problems: If a question requires certain items to be kept together (e.g., three friends must sit next to each other), 'glue' those items together and treat them as a single unit. Calculate the permutations for the larger group, and then multiply by the number of ways the 'glued' items can be arranged among themselves.
The '1 Minus' Shortcut for 'Not Together' Problems: To find the number of arrangements where certain items are not together, it's often easiest to find the total number of arrangements and subtract the number of arrangements where they are together (using the glue method).
The Matrix Method for Complex Probability: For problems involving two independent events with multiple outcomes (e.g., rolling two dice), drawing a simple grid or matrix can help you visualize all possible outcomes and systematically count the favorable ones.