Let C_r denote the binomial coefficient ^ n C_r . If _ r=1 ^ n r^2 ( C_r C_ r-1 ) = 286 , then the maximum value of the binomial coefficient C_r for this value of n is equal to

Step-by-step solution

We know that C_r C_ r-1 = n-r+1 r . Substituting this into the given summation: _ r=1 ^ n r^2 ( n-r+1 r ) = _ r=1 ^ n r(n-r+1) _ r=1 ^ n ( (n+1)r - r^2 ) = (n+1) _ r=1 ^ n r - _ r=1 ^ n r^2 (n+1) n(n+1) 2 - n(n+1)(2n+1) 6 n(n+1) 2 ( (n+1) - 2n+1 3 ) n(n+1) 2 ( 3n+3-2n-1 3 ) = n(n+1)(n+2) 6 We are given that this sum is equal to 286 . n(n+1)(n+2) 6 = 286 n(n+1)(n+2) = 1716 Since 11 12 13 = 1716 , we get n = 11 . For n = 11 , the maximum binomial coefficient is the middle term, which is ¹¹C₅ (or ¹¹C₆ ). ¹¹C₅ = 11 10