Class 11 Mathematics — Chapter 7: BINOMIAL THEOREM

What you'll learn

• ▸Pascal's Triangle
• ▸Binomial Coefficients as Combinations (ⁿCᵣ)
• ▸Binomial Theorem for Positive Integral Indices
• ▸Properties of Binomial Expansion
• ▸Special Cases of Binomial Expansion
• ▸Application in Numerical Evaluation
• ▸Advanced Applications: Comparison and Divisibility

Chapter Summary

--- PAGE 1 --- Chapter 7 BINOMIAL THEOREM ✠Mathematics is a most exact science and its conclusions are capable of absolute proofs. – C.P. STEINMETZ✠ 7.1 Introduction In earlier classes, we have learnt how to find the squares and cubes of binomials like a + b and a – b. Using them, we could evaluate the numerical values of numbers like (98)² = (100 – 2)², (999)³ = (1000 – 1)³, etc. However, for higher powers like (98)⁵, (101)⁶, etc., the calculations become difficult by using repeated multiplication. This difficulty was overcome by a theorem known as binomial theorem. It gives an easier way to expand (a + b)ⁿ, where n is an integer or a rational number. In this Chapter, we study binomial theorem for positive integral indices only. 7.2 Binomial Theorem for Positive Integral Indices Blaise Pascal Let us have a look at the following identities done earlier: (1623-1662) (a + b)⁰ = 1 a + b ≠ 0 (a +…

Practice Questions from this Chapter

Tap "Get Solution" on any question to ask our AI tutor.

1. Explain binomial theorem's real use. Get Solution →
2. Discover more Pascal's Triangle patterns. Get Solution →
3. Trace the history of binomial coefficients. Get Solution →
4. According to the text, how many terms are there in the expansion of (a + b)ⁿ? Get Solution →
5. The binomial theorem provides an easier way to expand what expression? Get Solution →
6. The triangular array of numbers used for binomial coefficients is named after which French mathematician? Get Solution →
7. In any term in the expansion of (a + b)ⁿ, what is the sum of the indices of 'a' and 'b'? Get Solution →
8. What are the coefficients ⁿCᵣ, which occur in the binomial theorem, known as? Get Solution →

Did you know?

• 💡 Pascal's Triangle numbers appear in the growth patterns of seashells and pinecones.
• 💡 The binomial theorem helps computer scientists design error-correcting codes for data.
• 💡 Every row in Pascal's Triangle sums up to a power of two, such as 1, 2, 4, 8.
• 💡 Blaise Pascal, the triangle's namesake, invented a mechanical calculator at just 19 years old.
• 💡 Combinations are essential for calculating odds in poker and other card games.

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