(x + y)ⁿ = Σ C(n, r) x^(n-r) y^r.

Solution

Maths sir · 4 steps

This is the binomial theorem itself: the powers of x fall while the powers of y rise, term by term.

1. Where terms come from. (x+y)ⁿ is n brackets multiplied together. Each term of the product picks y from r brackets and x from the other n-r, which gives xⁿ⁻ʳyʳ.
2. Count the ways. The r brackets that give y can be chosen in C(n, r) ways. So xⁿ⁻ʳyʳ appears exactly C(n, r) times.

   (x+y)ⁿ = Σᵣ₌₀ⁿ C(n, r) xⁿ⁻ʳ yʳ

3. Quick check. For n = 2: C(2, 0)x² + C(2, 1)xy + C(2, 2)y² = x² + 2xy + y², as expected.
4. So. With r running from 0 to n, the statement is the binomial theorem, so it is true.

The idea to keep. In (x+y)ⁿ the general term is Tᵣ₊₁ = C(n, r)xⁿ⁻ʳyʳ. The two exponents always add to n, and r runs from 0 to n, which gives n+1 terms.

Answer from the NTA final answer key. Solution by PrepWiser.