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Permutations and combinations are best understood when seen in action across varied conditions. A single situation can branch into multiple possibilities, each shaped by different restrictions or groupings. In the explanatory video for this article, followed by the written discussion, we explore eight very interesting scenarios that illustrate how outcomes change with each condition. Working through such variations is a valuable part of an efficient GMAT prep course, as it strengthens conceptual clarity and builds logical accuracy. By examining these cases together, you learn when to combine, when to arrange, and when to adjust for overlaps, making the topic intuitive.
With 5 boys and 5 girls, we simply have 10 people in a row.
Since all positions are open and order matters, the total number of arrangements is 10!
Treat the two boys as one unit.
Now, there are 9 entities in total.
These can be arranged in 9! ways.
The two boys can switch seats between themselves in 2! ways.
The total number of arrangements is 9! × 2!
Treat the three girls as one unit.
Now, there are 8 entities: one trio plus 7 individuals.
These can be arranged in 8! ways.
Within the trio, the girls can switch seats in 3! ways.
The total number of arrangements is 8! × 3!
Start with total arrangements (10!).
Subtract the cases where the two girls sit together.
Together, they form one unit plus 8 others, arranged in 9! ways.
They can swap places in 2! ways.
So, subtract 9! × 2! from 10! to get the answer.
The total number of arrangements is 10! – (9! × 2!)
Here, the boys form one block and the girls form another.
These two blocks can be swapped in 2! ways.
Within each block, the 5 members can be arranged in 5! ways.
The total number of arrangements is 2! × 5! × 5!
This can be either boy-girl-boy-girl or girl-boy-girl-boy.
That gives 2 cases (added, not multiplied).
In each case, boys can be arranged in 5! ways and girls in 5! ways.
The total number of arrangements is 5! × 5! × 2
Seat the boys first. The 5 boys can be arranged in 5! ways.
This creates 6 available slots between and around the boys.
The 5 girls must occupy 5 out of these 6 slots, in 6P5 ways.
The total number of arrangements is 5! × 6P5
Now, there are 6 girls and 5 boys.
First, seat the 6 girls in 6! ways.
This creates 7 slots around them.
The 5 boys must sit in 5 of these slots, in 7P5 ways.
The total number of arrangements is 6! × 7P5
These problems show the importance of identifying whether to treat groups as single units, subtract overlapping cases, or seat one group before placing another. Each restriction trains you to think structurally. Attempting enough full GMAT simulations is the best way to reinforce these variations under exam-like conditions.
| Case | Method | Final Expression |
|---|---|---|
| No restrictions (5 boys + 5 girls) | Arrange all 10 freely | 10! |
| Two particular boys always sit together | Treat them as 1 unit (9!); multiply by 2! for internal arrangement | 9! × 2! |
| Three particular girls always sit together | Treat them as 1 unit (8!); multiply by 3! for internal arrangement | 8! × 3! |
| Two particular girls never sit together | Total cases (10!) – cases where together (9! × 2!) | 10! – 9! × 2! |
| All boys together and all girls together | Treat boys as 1 unit and girls as 1 unit (2!); arrange within each group (5! × 5!) | 2! × 5! × 5! |
| Boys and girls alternate | Two possible seatings (×2); arrange boys in 5!, girls in 5! | 5! × 5! × 2 |
| No two girls sit together | Seat boys first (5!); girls placed in 6 slots (6P5) | 5! × 6P5 |
| No two boys sit together (with 6 girls) | Seat girls first (6!); boys placed in 7 slots (7P5) | 6! × 7P5 |
At due stage in your prep, utilize this free, full-length GMAT mock!
Permutations remind us that small changes in conditions can alter every outcome, while combinations show that essence sometimes matters more than sequence. In GMAT preparation, this means building discipline to spot subtle restrictions, practice systematically, and think structurally. In the MBA admission process, order shapes tasks such as scheduling exams, drafting essays, and securing recommendations, while substance defines the schools you choose and the values you highlight. Life too teaches us to notice when order carries weight and when the choice itself is the center. Wisdom lies in balancing both perspectives with awareness, patience, and clarity.
