Find the total number of terms in the expansion of (x + a)^51 - (x - a)^51 after simplification.

Notice  the first few  terms in the expansion of  both terms (where C1, C2, C3....   are  the terms on  the 51st row of Pascal's triangle)

(x +a)^51  =  C1 x^51  + C2x^50 a  + C3x^49a^2 + C4x^48 a^3 + C5x^47a^4  + C6 x^46a^5 +.......+.x^0*a^51

-(x - a)^51 = -C1x^51 + C2x^50a  -  C3x^49a^2   + C4x^48a^3  - C5x^47a^4  +  C6x^46a^5 +  ....+ x^0 * a^51

Adding  these  we  get

2C2x^50a  +  2C4x^48a^3  + C6x^46a^5 +  ....+ 2x^0a^51

The  number of  terms  is   [ 51 + 1]  / 2  =   26