If for 3 r 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals:

Step-by-step solution

Using the property ^ n C_ r = ^ n C_ n-r , the given expression can be written as: ³⁰C_ r + 3 ³⁰C_ r-1 + 3 ³⁰C_ r-2 + ³⁰C_ r-3 This expression represents the coefficient of x^r in the expansion of: (1+x)³⁰ + 3x(1+x)³⁰ + 3x^2(1+x)³⁰ + x^3(1+x)³⁰ Taking (1+x)³⁰ common, we get: (1+x)³⁰(1 + 3x + 3x^2 + x^3) = (1+x)³⁰(1+x)^3 = (1+x)³³ The coefficient of x^r in (1+x)³³ is ³³C_ r . Comparing this with ^ m C_ r , we get m = 33 . Answer: 33