If the second, third and fourth terms in the expansion of `(x+y)^(n)` be 135, 30 and 10/3 respectively, then

Options

A. ` n= 7`
B. n= 5
C. n = 6
D. none of these

Answer

B

Explanation

We have,
`T_(2) = 135 rArr ""^(n)C_(3) x^(n-2) y^(2) = 135`
and, ` T_(3)= (10)/(3) rArr ""^(n)C_(2) x^(n-2) y^(2) = 30`
and ,
`T_(4) = (10)/(3) rArr ""^(n)C_(3) x^(n-3) y^(3) = (10)/(3)`
Now, `(T_(2)xxT_(4))/(T_(3)""^(2))= (135xx(10)/(3))/(30)^(2)`
`rArr ((""^(n)C_(1) x^(n - 1)y^(1))xx(""^(n)C_(1) x^(n - 3)y^(3)))/((""^(n)C_(2) x^(n - 2)y^(2)))=(135xx(10)/(3))/((30)^(2))`
`rArr (""^(n)C_(1) xx""^(n)C_(3))/(""^(n)C_(2))^(2) = (135xx10)/(2700)`
`rArr (nxxn(n - 1) (n-2)//3!)/((n(n-1)//2!)^(2)) = (1)/(2)`
`rArr (n-2)/(n-1)xx(2)/(3) = (1)/(2) rArr 4n - 8 = 3n - 3 rArr n = 5`