9 min read
MIXED COUNTING AND nCr IDENTITIES
A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
Choose Your Level
MIXED COUNTING AND nCr IDENTITIES
What is it?
A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
In simple words
- A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys.
- Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out').
- Meaning: split on a condition, add the disjoint piles.
- At least one is not nC1. It is a sum of cases or a complement.
Important terms
MIXED COUNTING AND nCr IDENTITIES: A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
1. AT LEAST / AT MOST: At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
2. PASCAL AND PATHS: A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.
What you should learn
- State the main ideas and notation used in mixed counting and ncr identities.
- Apply the rules of mixed counting and ncr identities in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- All - none
- Add k=0 to k=2
- Product inside a case if two types
AT LEAST / AT MOST
At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
PASCAL AND PATHS
A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.
Recommended for Class 11
Full school-level explanation
A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
Cases, then Pascal:
WHAT — MIXED COUNTING AND nCr IDENTITIES
A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
RULE — 1. AT LEAST / AT MOST
At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
METHOD — 2. PASCAL AND PATHS
A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.
WORKED EXAMPLE
Worked: committees of 3 from 5 people, at least one of A or B (idea: complement). total 5C3 = 10 with neither A nor B: 3C3 = 1 so 10-1 = 9 have A or B or both
AT LEAST / AT MOST
At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
Remember
- complement is faster often
Examples
- All - none
- Add k=0 to k=2
- Product inside a case if two types
PASCAL AND PATHS
A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.
Remember
- in or out
Examples
- (n-1)C(r-1) + (n-1)Cr = nCr
- Triangle of nCr
- Check small numbers
How to answer a mixed counting and ncr identities question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- at least: cases or complement
- Pascal: nCr + nC(r-1) = (n+1)Cr
- 9
- not just nC1
- complement is faster often
- in or out
- at least: cases or complement
- Pascal: nCr + nC(r-1) = (n+1)Cr
How the ideas connect
Worked: committees of 3 from 5 people, at least one of A or B (idea: complement). total 5C3 = 10 with neither A nor B: 3C3 = 1 so 10-1 = 9 have A or B or both
Deeper understanding
Go beyond memorising MIXED COUNTING AND nCr IDENTITIES. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: committees of 3 from 5 people, at least one of A or B (idea: complement). total 5C3 = 10 with neither A nor B: 3C3 = 1 so 10-1 = 9 have A or B or both
Common mistakes
Exam focus
Core Idea
1. AT LEAST / AT MOST
2. PASCAL AND PATHS
Worked Example
Exam focus
Quick revision
- at least: cases or complement
- Pascal: nCr + nC(r-1) = (n+1)Cr
- 9
- not just nC1
- MIXED COUNTING AND nCr IDENTITIES: A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement
- 1. AT LEAST / AT MOST: At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
At least one is not nC1. It is a sum of cases or a complement.
For long answers on MIXED COUNTING AND nCr IDENTITIES: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- at least: cases or complement
- Pascal: nCr + nC(r-1) = (n+1)Cr
- 9
- not just nC1
Important formulas
- at least: cases or complement
- Pascal: nCr + nC(r-1) = (n+1)Cr
- 9
Common exam traps
- At least one is not nC1. It is a sum of cases or a complement.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
At least one is not nC1. It is a sum of cases or a complement.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
At least one is not nC1. It is a sum of cases or a complement.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
at least: cases or complement
Pascal: nCr + nC(r-1) = (n+1)Cr
9
not just nC1
A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (s
Pascal: nCr + nC(r-1) = (n+1)Cr
- Definition
- A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (s
- Formula
- Pascal: nCr + nC(r-1) = (n+1)Cr
- Example
- All - none
- Critical warning
- At least one is not nC1. It is a sum of cases or a complement.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define MIXED COUNTING AND nCr IDENTITIES.
Explain
Q2. Explain: AT LEAST / AT MOST.
Example
Q3. Give one clear example of MIXED COUNTING AND nCr IDENTITIES and say why it fits.
Mistake
Q4. State one common mistake students make about MIXED COUNTING AND nCr IDENTITIES.
Exam note
Q5. Write a short exam note on MIXED COUNTING AND nCr IDENTITIES: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.