The value of r for which `.^(20)C_(r)xx.^(20)C_(0)+.^(20)C_(r-1)xx.^(20)C_(1)+.^(20)C_(r-2)xx.^(20)C_(2)+...+.^(20)C_(0)xx.^(20)C_(r)` is maximum is

Options

A. 15
B. 10
C. 11
D. 20

Explanation

We know that , `(1+x)^(20) =""^(20)C_(0)+""^(20)C_(1)x+""^(20)C_(2)x^(2) +...+""^(20)C_(r-1)x^(r-1)+""^(20)C_(r)x^(r)+...+""^(20)C_(20)x^(20)`
`:. (1+x)^(20)*(1+x)^(20)=(""^(20)C_(0)+""^(20)C_(1)x+""^(20)C_(2)x_(2) +...+""^(20)C_(r-1)x^(r-1)+""^(20)C_(r)x^(r)+...+""^(20)C_(20)x^(20) ) xx (""^(20)C_(0)+""^(20)C_(1)x+...+""^(20)C_(r-1)x^(r-1)+""^(20)C_(r)C^(r)+...+""^(20)C_(20)x^(20)) `
` implies (1+x)^(40)=(""^(20)C_(0)*""^(20)C_(r)+""^(20)C_(1)*""^(20)C_(r-1)...""^(20)C_(r)""^(20)C_(0))x^(r)+....`
On comparing the coefficient of `x^(r)` of both sides, we get
`""^(20)C_(0)*""^(20)C_(r)+""^(20)C_(1)""^(20)C_(r-1)+...+""^(20)C_(r)""^(20)C_(0)=""^(40)C_(r)`
The maximum value of `""^(40)C_(r)` is possible only when r=20 `" "` [ `:' ""^(n)C_(n//2)` is maximum when n is even ]
Thus , required value of r is 20.