Question

What is the binomial expansion of (x + 2y)7?

Answer 1

Thus, the required "option 4)"  is correct.

Step-by-step explanation:

We find,

$(x+2y)^7$

To find, the binomial expansion of of $(x+2y)^7$ = ?

We know that,

$(x+y)^{n} =^nC_0x^n+^nC_1x^{n-1}y+nC_2x^{n-2}y^2+nC_3x^{n-3}y^3+...+^nC_ny^n$

∴ $(x+2y)^7$

Here, n = 7, x = x and y = 2y

$(x+2y)^7$= $^7C_0x^7+^7C_1x^{7-1}(2y)+^7C_2x^{7-2}(2y)^2+^7C_3x^{7-3}(2y)^3+^7C_4x^{7-4}(2y)^4+^7C_5x^{7-5}(2y)^5+^7C_6x(2y)^6+(2y)^7$$=x^7+7x^{6(2y)+21x^{5}4y^2+35x^{4}8y^3+^35x^{3}16y^4+21x^{2}32y^5+7x64y^6+128y^7$

$=x^7+14x^{6}y+84x^{5}y^2+280x^{4}y^3+560x^{3}y^4+672x^{2}y^5+448xy^6+128y^7$

∴ The binomial expansion of of $(x+2y)^7$

$=x^7+14x^{6}y+84x^{5}y^2+280x^{4}y^3+560x^{3}y^4+672x^{2}y^5+448xy^6+128y^7$

Thus, the required "option 4)"  is correct.