Quantitative Aptitude / Modern mathematics

Binomial Concepts

Binomial Concepts: concepts, worked examples and practice.

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Start with the basics

Binomial Concepts, explained simply

A binomial has two terms, such as a + b. Expanding a product means multiplying each term carefully.

Try a small example

Expand (x + 2)(x + 3).

Show the steps

Multiply x by x and by 3: x² + 3x.
Multiply 2 by x and by 3: 2x + 6.
Combine like terms: x² + 5x + 6.

Remember: Multiply every term in the first bracket by every term in the second.

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Binomial Expansion

The binomial theorem expands (a+b)^n for a non-negative integer n. A term selects a from some factors and b from the rest.

Worked example

(x+2)^3 = x^3 + 6x^2 + 12x + 8. The coefficient 6 comes from choosing one 2 among three factors: 3 x 2.

Binomial Coefficients

The coefficient nCr counts the ways to select r factors contributing the second term. Coefficients are symmetric: nCr = nC(n-r).

Worked example

In (1+x)^5, the coefficient of x^2 is 5C2 = 10. The sum of all coefficients is found by putting x=1, giving 2^5=32.

Exam focus

CAT: Quantitative Aptitude

Start with arithmetic and algebra. Check domain restrictions and use estimation before long calculations; specialised topics are supporting preparation.

XAT: Quantitative Aptitude and Data Interpretation

Combine common quantitative skills with accurate interpretation of unfamiliar data. Recheck units and assumptions.

SNAP: Quantitative, Data Interpretation and Data Sufficiency

Choose a fast method only after identifying the base and units. In sufficiency questions, test uniqueness rather than calculating unnecessarily.

Check your understanding

Explain the idea in your own words before opening the answer. Then apply it to the example.

Binomial Expansion

Explain the main rule in binomial expansion, then apply it to a small example. State the assumption that makes your reasoning valid.

Review the note

Check your answer

The binomial theorem expands (a+b)^n for a non-negative integer n. A term selects a from some factors and b from the rest.

Worked example: (x+2)^3 = x^3 + 6x^2 + 12x + 8. The coefficient 6 comes from choosing one 2 among three factors: 3 x 2.

Binomial Coefficients

Explain the main rule in binomial coefficients, then apply it to a small example. State the assumption that makes your reasoning valid.

Review the note

Check your answer

The coefficient nCr counts the ways to select r factors contributing the second term. Coefficients are symmetric: nCr = nC(n-r).

Worked example: In (1+x)^5, the coefficient of x^2 is 5C2 = 10. The sum of all coefficients is found by putting x=1, giving 2^5=32.

Practice questions

Bharatiya Pathshala

Opening preparation notes...

Binomial Concepts, explained simply

Try a small example

Binomial Expansion

Worked example

Binomial Coefficients

Worked example

CAT: Quantitative Aptitude

XAT: Quantitative Aptitude and Data Interpretation

SNAP: Quantitative, Data Interpretation and Data Sufficiency

Check your understanding

Binomial Expansion

Binomial Coefficients

Binomial Concepts, explained simply

Try a small example

Binomial Concepts: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Formulas to remember

Binomial Expansion

Worked example

Binomial Coefficients

Worked example

Application and strategy

Worked application

Common traps and strategy

CAT: Quantitative Aptitude

XAT: Quantitative Aptitude and Data Interpretation

SNAP: Quantitative, Data Interpretation and Data Sufficiency

Check your understanding

Binomial Expansion

Binomial Coefficients

Binomial Concepts: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Formulas to remember

Application and strategy

Worked application

Common traps and strategy