The number of non zero terms in the expansion of \(\rm (1+3 \sqrt{2} x)^{9}+(1-3 \sqrt{2} x)^{9}\) is

Calculation:

Given:

(1 + 3√2x)⁹ + (1 - 3√2x)⁹

Applying binomial theorem for expansion of the following

= 1 + (3√2x)(\({^9C_ 1}\)) + (3√2x)²(\({^9 C_ 2}\)) + ... + (\({^9 C_ 9}\)(3√2x)⁹)

+ 1 - (3√2x)(\({^9C_ 1}\)) + (3√2x)²(\({^9 C_ 2}\)) - ... - (\({^9 C_ 9}\)(3√2x)⁹)

= 2[1 + (3√2x)²(\({^9 C_ 2}\)) + (3√2x)⁴(\({^9 C_ 4}\)) + ... + (3√2x)⁸(\({^9C_ 8}\))]

The powers of 3√2x are in A.P.

0, 2, 4, 6, 8

Hence, including the term 1 or x⁰ there are (9+1)/2 terms.

Hence, there are 5 terms.

Hence option 4 is correct