Invest 30 seconds...
...for what may lead to a life altering association!
Help Line
- +91.8800.2828.00 (IND)
- 1030-1830 Hrs IST, Mon-Sat
- support@expertsglobal.com
Practice combinations through eight committee selection cases: no restriction; exact 2 boys and 3 girls; at least one boy; at most four boys; fixed inclusion; banned members; conditional pair (if Jack then Lily); together-or-neither. Use nCr, casework, and complementary counting to structure selections.
Combinations form the backbone of many counting questions because they capture situations where order does not matter. This article, along with the explanatory video, presents a set of eight illustrative scenarios that show how subtle conditions – such as inclusions, exclusions, or minimum requirements – change the count in important ways. Seeing these variations side by side brings out the real logic of combinatorics and highlights when complementary counting or case division is more efficient. Working through such examples is an integral part of sound GMAT prep, since clarity on structure directly improves accuracy. Reinforcing these ideas through adequate GMAT mocks further ensures that the reasoning becomes second nature under real test conditions.
Committee-based problems are one of the best ways to understand the practical application of combinations. Let us walk through different cases step by step, each teaching a distinct principle.
Since this is a committee, order does not matter.
If a committee of 5 is to be formed out of 11 individuals, the answer is 11C5.
Out of 5 boys, select 2, and out of 6 girls, select 3.
These selections happen simultaneously, so the answer is 5C2 × 6C3.
We count all possibilities (11C5) and subtract the case where no boy is chosen (6C5).
Thus, the answer is 11C5 – 6C5.
While it can also be solved by adding cases (1 boy, 4 girls; 2 boys, 3 girls, etc.), the subtraction method is more efficient.
This excludes the case where all five members are boys.
The answer is 11C5 – 5C5.
If one person is already chosen, the committee reduces to selecting 4 from the remaining 10.
The answer is 10C4.
The pool of candidates reduces from 11 to 9.
The answer is 9C5.
Here, there are two scenarios:
(a) If Jack and Lily are both selected, 3 more members are chosen from the remaining 9: 9C3.
(b) If Jack is not selected, the committee of 5 comes from the remaining 10: 10C5.
The answer is 9C3 + 10C5.
When both are included, the remaining 3 are chosen from 9: 9C3.
When both are excluded, all 5 are chosen from 9: 9C5.
The answer is 9C3 + 9C5.
If you could solve all of these, you have a solid grasp of the basics of combinations. These cases highlight efficiency in problem-solving, showing when subtraction saves time and when breaking into cases is necessary.
| Case | Condition | Expression | Explanation |
|---|---|---|---|
| 1 | No restriction | 11C5 | Select any 5 out of 11; order does not matter. |
| 2 | Exactly 2 boys and 3 girls | 5C2 × 6C3 | Choose 2 boys from 5 and 3 girls from 6 simultaneously. |
| 3 | At least 1 boy | 11C5 – 6C5 | Subtract the all-girls case from total selections. |
| 4 | At most 4 boys | 11C5 – 5C5 | Exclude the all-boys case. |
| 5 | A particular girl always included | 10C4 | One member fixed; choose remaining 4 from 10. |
| 6 | Two particular boys never included | 9C5 | Exclude both boys; choose 5 from the remaining 9. |
| 7 | Jack optional, Lily may still be chosen | 9C3 + 10C5 | Case 1: Jack and Lily included → 3 from 9. Case 2: Jack excluded → 5 from 10. |
| 8 | Jack and Lily together or both absent | 9C3 + 9C5 | Case 1: Both included → 3 from 9. Case 2: Both excluded → 5 from 9. |
Combinations illustrate how selections shift when restrictions are applied, whether through fixed members, exclusions, or minimum and maximum conditions. The main insight is that order does not matter, and efficiency often comes from using complementary counting or case division. Practicing these variations builds a habit of logical structuring rather than relying on memorized shortcuts. Working through examples in a focused manner strengthens accuracy and confidence, while attempting full-length GMAT simulation may help confirm whether this understanding translates smoothly into timed problem solving under exam-style conditions.
Combinations remind us that sometimes the power of choice matters more than the order in which choices occur. In GMAT prep, success often depends on identifying the essential structure of a problem rather than being lost in sequence. In MBA admissions, the same wisdom applies when evaluating experiences and strengths to highlight, independent of chronology. Life, too, presents moments where presence outweighs order, where being part of the picture matters more than when or how. Clarity comes from recognizing when essence is enough, allowing decisions to be made with confidence and purpose.
