Let S(r) = ²⁰C_ 20-r + 4 ²⁰C_ 21-r + 6 ²⁰C_ 22-r + 4 ²⁰C_ 23-r + ²⁰C_ 24-r . If the ratio S(r) : S(r+1) = 2:3 , then the value of r is:

Step-by-step solution

Using the property ^ n C_ r = ^ n C_ n-r , the expression for S(r) can be rewritten as: S(r) = ²⁰C_ r + 4 ²⁰C_ r-1 + 6 ²⁰C_ r-2 + 4 ²⁰C_ r-3 + ²⁰C_ r-4 This sum represents the coefficient of x^r in the expansion of: (1+x)²⁰(1 + 4x + 6x^2 + 4x^3 + x^4) Recognizing the second factor as the binomial expansion of (1+x)^4 , we get: (1+x)²⁰(1+x)^4 = (1+x)²⁴ Thus, the coefficient of x^r is S(r) = ²⁴C_ r . We are given the ratio S(r) S(r+1) = 2 3 . Substituting S(r) = ²⁴C_ r and S(r+1) = ²⁴C_ r+1 : ²⁴C_ r ²⁴C_ r+1 = 24! r!