A combination is a selection of items where the order doesn't matter.
Use combinations when the order of selection does not matter.
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Combination
Combination is a fundamental concept in mathematics that refers to selecting items from a group, where the order does not matter. It is widely used in probability, statistics, and real-life problem-solving. The combination formula helps determine how many ways a subset can be chosen from a larger set, with or without repetition. Unlike permutations, combinations focus solely on the selection of objects, not their arrangement. Mastering combinations is essential for competitive exams like JEE and other advanced-level assessments.
1.0Combination Meaning
The term combination refers to the number of ways in which a group of items can be chosen from a larger set without considering the order of the items.
For example, choosing 2 fruits from a basket of {apple, banana, cherry} results in the combinations:
{apple, banana}
{apple, cherry}
{banana, cherry}
Note: {apple, banana} and {banana, apple} are considered the same combination.
2.0Combination Formula in Maths
Standard Combination Formula
If you are choosing r items from n, the number of combinations is given by: nCr=r!(n−r)!n!
Where:
n = total number of items
r = number of items chosen
! = factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)
Combination with Repetition Formula
If repetitions are allowed, the formula becomes: nCrwith repitition=r!(n−1)!(n+r−1)!
This is used when you're selecting items and each item can appear more than once.
Related Video:
3.0Solved Examples on Combination
Example 1: From a group of 7 students, how many ways can a committee of 3 students be formed?
Solution:7C3=3!(7−3)!7!=P3×2×17×6×5=35
Example 2: How many ways can you choose 3 fruits from 5 types, if repetition is allowed?
Solution:5+3−1C3=7C3=3!(7−3)!7!=35
Example 3: A team of 4 players is to be formed from 6 boys and 5 girls. If the team must include at least 2 girls, how many such teams can be formed?
Solution:
Break into cases:
2 girls, 2 boys: 5C2×6C2=10×15=150
3 girls, 1 boy: 5C3×6C1=10×6=60
4 girls: 5C4=5
Total = 150 + 60 + 5 = 215
Example 4: A committee of 5 is to be formed from 8 men and 4 women. How many ways can it be done if the committee must contain at least 2 women?
Solution: We consider all cases where number of women ≥ 2:
Case 1: 2 women, 3 men:=4C2×8C3=6×56=336
Case 2: 3 women, 2 men: =4C3×8C2=4×28=112
Case 3: 4 women, 1 man: =4C4×8C1=1×8=8
Total = 336 + 112 + 8 = 456 ways
Example 5: In how many ways can 4 identical balls be placed into 3 distinct boxes, where a box can hold any number of balls?
Solution: This is a combination with repetition problem:
We must solve for non-negative integers x1+x2+x3=4
Formula: =n+r−1Cr=3+4−1C4=6C4=15
Example 6: Prove using combination logic that: ∑k=0nnCk=2n
Solution:
This identity tells us the sum of all combinations of selecting 0 to n items from n items is 2n.
Total number of subsets of a set with n elements = 2n
nC0+nC1+....+nCn=2n
Proved
Example 7: How many 4-letter words can be formed using the letters of the word ‘ENGINEERING’ such that all vowels are included only once?
Solution:
Letters in 'ENGINEERING':
E, N, G, I, N, E, E, R, I, N, G
Vowels = E, I (E ×3, I ×2) → usable once each → choose 2 vowels
Consonants available: N, G, R (multiple copies) → choose 2 distinct consonants
Let’s assume vowels used once only: 2 vowels from {E, I} (max possible 2)
→ We must choose 2 consonants from N, G, R
Ways to choose 2 vowels = 2C2=1
Ways to choose 2 consonants (Distinct) =3C2=3
Ways to arrange 4 chosen letters = 4!=24
Total = 1 × 3 × 24 = 72
Answer = 72 ways
Example 8: In how many ways can 5 identical balls be distributed among 3 boys so that each boy gets at least 1 ball?
Solution:
We solve: x1+x2+x3=5 where, xi≥1
Convert to standard form: Let yi=xi−1⇒y1+y2+y3=2
No of Solutions = 2+3−1C2=4C2=6
Answer = 6 ways
4.0Applications of Combinations
Probability theory: Finding the number of possible outcomes.
Statistics: Selecting samples from populations.
Algebra: Solving binomial expansion problems.
Real life: Group selections, team formation, lottery systems.
5.0Key Differences: Permutation vs Combination
Feature
Permutation
Combination
Order Matters?
Yes
No
Formula
(n−r)!n!
r!(n−r)!n!
Example
Arranging books
Choosing books
6.0Sample Question on Combinations
Q1. What is the combination formula with repetition?