Find the number of rational terms and also find
the sum of rational terms in ` (sqrt(2) + root(3)(3) + root(6)(5) )^(10)`

Options

A. 12632
B. 1260
C. 126
D. None of these

Answer

a

Explanation

General term of given expansion
` = (10!)/(alpha!beta ! gamma!)2^(alpha//2) *3^(beta//3)*5^(gamma//6)` ...(i)
`alpha, beta , gamma ` satisfyting two following property
`0 le alpha, beta ,gamma le 10, alpha + beta + gamma = 10`
` therefore alpha= 0,2,4,6,8,10, beta = 0,3,6,gamma =0,6 `
Hence , possible pairs of ` (alpha , beta , gamma ) = (4,6,0),(4,0,6),(10,0,0) `
` therefore ` There exists three rational terms .
So, sum of rational terms
`= (10!)/(4!6!)*2^(2) *3^(2) + (10!)/(4!6!) 2^(2)*5^(1) + (10!)/(10!)2^(5) = 12632` .