Find the coefficient of y3 in (7y2 - \(\frac{2}{y})^{12}\)

1.	Find the coefficient of y3 in (7y2 - \(\frac{2}{y})^{12}\)
Answer» Here x =7y² , a = - \(\frac{2}{y}\) and n = 12 T_r+1 = ¹²C_r (7Y²)^12-r .(\(\frac{-2}{y}\))^r = ¹²C_r.7^12-r .y^24-2r. y^-r (-2)^r T_r+1 = ¹²C_r.7^12-r .(-2)^r .y^24-3r To find the coefficient of y³ equate the power of y or 3 i.e., 24 – 3r = 3 ⇒ 21 = 3r ⇒ r= 7 ∴T₇₊₁ = ¹²C₇ .7^12-7 . (-2)⁷ y³ = ^-12C₇ .7⁵ 2⁷ . y3 ∴Coefficient of y³ is ^-12C₇.7⁵ .2⁷

Answer»

Here x =7y² , a = - \(\frac{2}{y}\) and n = 12

T_r+1 = ¹²C_r (7Y²)^12-r .(\(\frac{-2}{y}\))^r

= ¹²C_r.7^12-r .y^24-2r. y^-r (-2)^r

T_r+1 = ¹²C_r.7^12-r .(-2)^r .y^24-3r

To find the coefficient of y³ equate the power of y or 3

i.e., 24 – 3r = 3 ⇒ 21 = 3r ⇒ r= 7

∴T₇₊₁ = ¹²C₇ .7^12-7 . (-2)⁷ y³

= ^-12C₇ .7⁵ 2⁷ . y3

∴Coefficient of y³ is ^-12C₇.7⁵ .2⁷

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