find the number of terms in the expansion
( 3x + y )^8 - ( 3x-y )^8

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} {\left(3\mathrm{x}+\mathrm{y}\right)}^8-{\left(3\mathrm{x}-\mathrm{y}\right)}^8 \\ =\left({\left(3\mathrm{x}\right)}^8+8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}+8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2+....8_{{\mathrm{C}}_8}{\mathrm{y}}^8\right)-\left({\left(3\mathrm{x}\right)}^8-8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}+8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2+.....+.8_{{\mathrm{C}}_8}{\mathrm{y}}^8\right) \\ =\mathrm{all} \mathrm{the} \mathrm{even} \mathrm{power} \mathrm{terms} \mathrm{will} \mathrm{be} \mathrm{cancelled} \\ ={\left(3\mathrm{x}\right)}^8+8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}+8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2+....+8_{{\mathrm{C}}_8}{\mathrm{y}}^8-{\left(3\mathrm{x}\right)}^8+8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}-8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2+.....-.8_{{\mathrm{C}}_8}{\mathrm{y}}^8 \\ ={\left(3\mathrm{x}\right)}^8-{\left(3\mathrm{x}\right)}^8+8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}+8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2-8_{{\mathrm{C}}_2}{\left(3\mathrm{x}\right)}^6.{\mathrm{y}}^2+8_{{\mathrm{C}}_3}{\left(3\mathrm{x}\right)}^5.{\mathrm{y}}^3+8_{{\mathrm{C}}_4}{\left(3\mathrm{x}\right)}^4.{\mathrm{y}}^4-8_{{\mathrm{C}}_4}{\left(3\mathrm{x}\right)}^4.{\mathrm{y}}^4+8_{{\mathrm{C}}_5}{\left(3\mathrm{x}\right)}^3.{\mathrm{y}}^5+8_{{\mathrm{C}}_6}{\left(3\mathrm{x}\right)}^2.{\mathrm{y}}^6-8_{{\mathrm{C}}_6}{\left(3\mathrm{x}\right)}^2.{\mathrm{y}}^6+8_{{\mathrm{C}}_7}{\left(3\mathrm{x}\right)}^1.{\mathrm{y}}^7+8_{{\mathrm{C}}_8}{\mathrm{y}}^8-8_{{\mathrm{C}}_8}{\mathrm{y}}^8 \\ =2\left(8_{{\mathrm{C}}_1}{\left(3\mathrm{x}\right)}^7.\mathrm{y}+8_{{\mathrm{C}}_3}{\left(3\mathrm{x}\right)}^5.{\mathrm{y}}^3+8_{{\mathrm{C}}_5}{\left(3\mathrm{x}\right)}^3.{\mathrm{y}}^5+8_{{\mathrm{C}}_7}{\left(3\mathrm{x}\right)}^1.{\mathrm{y}}^7\right) \\ \mathrm{hence} \mathrm{it} \mathrm{will} \mathrm{have}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{terms} \end{gathered}$$



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