The number of terms in (x + y + z)² is x.

Solution

Maths sir · 3 steps

Count distinct terms by counting ways to split the power 2 among x, y and z.

1. Shape of a term. Every term in the expansion has the form xᵃ yᵇ zᶜ with a + b + c = 2 and each power a whole number ≥ 0. Each different triple (a, b, c) gives a different term. Here the power is n = 2 and there are r = 3 variables.

   Number of terms = C(n + r - 1, r - 1) = C(2 + 3 - 1, 3 - 1) = C(4, 2) = 6

2. Check by expanding. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. The three squares and the three cross products are all different, giving 6 terms.
3. So. The expansion has 6 terms.

The common slip: 9. Counting xy and yx as separate terms gives 9, not 6. They are like terms and combine into 2xy.

The idea to keep. In (x₁ + x₂ + … + xᵣ)ⁿ the number of distinct terms is C(n + r - 1, r - 1). It is the number of ways to share the power n among the r variables.

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