If for all valid integers r , the identity ²²C_ 22-r + a ²²C_ 23-r + b ²²C_ 24-r + c ²²C_ 25-r + ²²C_ 26-r = ²⁶C_ r holds true, then the value of a+b+c is:

Step-by-step solution

Using the property ^ n C_ r = ^ n C_ n-r , the given expression on the left hand side can be written as: ²²C_ r + a ²²C_ r-1 + b ²²C_ r-2 + c ²²C_ r-3 + ²²C_ r-4 This expression represents the coefficient of x^r in the expansion of: (1+x)²² + ax(1+x)²² + bx^2(1+x)²² + cx^3(1+x)²² + x^4(1+x)²² = (1+x)²²(1 + ax + bx^2 + cx^3 + x^4) The right hand side is ²⁶C_ r , which is the coefficient of x^r in (1+x)²⁶ . Equating the polynomials, we get: (1+x)²²(1 + ax + bx^2 + cx^3 + x^4) = (1+x)²⁶ 1 + ax + bx^2 + cx^3 + x^4 = (1