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Permutation and combination link counting to probability. Probability equals favorable cases over total cases. Example: five books are shuffled; what is the chance physics and math end up together? Favorable = 4! × 2!, total = 5!, so probability = 4!×2!/5! = 2/5.
Permutation and combination problems in GMAT prep are especially engaging as they reveal how structured counting leads directly to probability. In P&C-based probability, the numerator is the favorable cases and the denominator the total cases. By defining all possible arrangements first and then applying restrictions, every case becomes clear. Sometimes the challenge involves grouping, sometimes avoiding overlap, and sometimes calculating complements, but each variation strengthens logical skills. This disciplined reasoning sharpens accuracy and builds confidence, becoming second nature with consistent practice in a GMAT test series.
Questions that merge probability with permutations and combinations require a layered approach, since they involve both calculating total possibilities and applying specific restrictions. The first step is always to establish the unrestricted denominator, which captures all potential arrangements. Once this is clear, different tools such as grouping, gap placement, or the complement principle allow you to account for conditions like “together” or “never together.” These strategies convert complex scenarios into manageable steps. Let us now examine the example from the video, which presents four distinct parts.
Question: 5 boys and 5 girls have to be seated in a row. Find the probability that…
We now examine this multi-part seating arrangement problem involving 10 people in a row, five boys and five girls.
The denominator in every case is the total number of unrestricted arrangements, which is 10!.
We can treat the boys as one block and the girls as another block.
Thus, there are 2 groups, which can be arranged in 2! ways.
Inside each block, the 5 boys can be arranged in 5! ways, and the 5 girls in 5! ways.
The denominator, representing all possible seating arrangements, is 10!.
Thus,
Favorable arrangements = 2! × 5! × 5!
Total arrangement = 10!
Probability = (2! × 5! × 5!) / 10!
Therefore, the required probability = (2! x 5! x 5!) ÷10!
Here, it is better to place the unrestricted group first.
The 5 girls can be arranged in 5! ways.
This creates 6 available gaps (including the two ends) where the 5 boys must sit.
The boys can take any 5 of the 6 positions, arranged in 6P5 ways, and then ordered among themselves in 5! ways.
Thus,
Favorable arrangements = 5! × 6P5
Total arrangements = 10!
Therefore, the required probability = (5! × 6P5) ÷ 10!
Treat the two specified boys as a single unit.
Now, instead of 10 students, we have 9 elements: the combined block, the 3 remaining boys, and the 5 girls.
These 9 elements can be arranged in 9! ways.
Within the block, the two boys can be swapped in 2! ways.
The denominator remains 10!.
Thus,
Favorable arrangements = 9! × 2!
Total arrangements = 10!
Therefore, the required probability = ((9! × 2!)) ÷ 10!
This case is solved using the complement method.
The probability of the two girls being together is the same as in Case 3.
Hence, the probability of them never being together is 1 minus that probability.
Thus,
Favorable arrangements = 10! − (9! × 2!)
Total arrangements = 10!
Therefore, the required probability = [10! − (9! × 2!)] ÷ 10!
These problems demonstrate that clarity of thought develops when restrictions are handled systematically rather than by listing every possible case. Begin by fixing the total possibilities, then bring in grouping, gaps, or complements as required. This approach simplifies complex scenarios into manageable steps. Practicing such structured reasoning through full-length GMAT simulation allows you to build accuracy and speed in exam-like conditions, ensuring that even advanced probability and permutation problems can be solved with confidence and efficiency.
Probability and permutation questions remind us that order and restriction shape outcomes in powerful ways. On the GMAT, success comes from structuring the problem, not from listing endless cases. The MBA admissions process teaches a similar truth: presenting experiences with clarity and balance makes a story persuasive. Life too unfolds as a mix of freedom and constraint, where choices are framed by context. Those who pause to structure their approach, instead of reacting randomly, find the path to clarity. True progress often lies in carefully arranging possibilities until order emerges from complexity.
