The value of Σᵣ₌₀⁶ ( ⁶Cᵣ . ⁶C₆₋ᵣ ) is equal to :

Solution

Maths sir · 4 steps

Each product ⁶Cᵣ · ⁶C₆₋ᵣ is one way to build x⁶ from two copies of (1+x)⁶.

1. Two expansions. Multiply (1+x)⁶ by (1+x)⁶. To get x⁶, take xʳ from the first factor and x⁶⁻ʳ from the second. Their coefficients are ⁶Cᵣ and ⁶C₆₋ᵣ.
2. Add every split. Adding over r = 0 to 6 counts every way to make x⁶. So the sum is the coefficient of x⁶ in (1+x)¹², which is ¹²C₆.

   Σᵣ₌₀⁶ ⁶Cᵣ ⁶C₆₋ᵣ = ¹²C₆ = 12!/(6! 6!) = 924

3. Quick check. Since ⁶C₆₋ᵣ = ⁶Cᵣ, each term is (⁶Cᵣ)². That gives 1 + 36 + 225 + 400 + 225 + 36 + 1 = 924.
4. So. The sum equals 924, which is option (a).

The idea to keep. When two binomial coefficients have lower indices adding to a fixed number, compare coefficients in (1+x)ᵐ(1+x)ⁿ. You get ᵐ⁺ⁿCₖ, which is Vandermonde's identity.

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