No. of distinct terms in (1+x^2+x^4)^20

As, all the terms are positive, no terms will get cancelled out.

Let $$x^2$$ = y. Then the expression becomes, $$(1 + y + y^2)^{20}$$

The maximum power of y in the above expression is 40.

If a is the power of 1, b is the power of $$y$$ and c is the power of $$y^2$$, then a + b + c = 20

And the power of y becomes b + 2c.

Now observe that all the numbers from 0 to 40 can be expressed as b + 2c , where a + b + c = 20

To prove that, we use induction. 40 can be expresed as 40 = 0 + 2 * 20, 0 + 0 + 20 = 20,

Let say X = b + 2c and a + b + c = 20,

then X -1 = b+1 + 2(c -1) , if c  > 0 and (a + (b+1) + (c -1) ) = 20

or X -1 = b -1 + 2c, if c = 0 and (a -1 + b -1 + c ) = 20.

Hence all the powers from 0 to 40 can be expressed.

Hence the total number of terms is 41

Thank u very much