Question

One or more options can be correct:
Find the coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

• A. ${}^{12}{C}_2$
• B. ${}^{12}{C}_2{-}^5{C}_2{-}^4{C}_2{-}^3{C}_3$
• C. $12C_2{-}^5{C}_3{-}^4{C}_2{-}^3{C}_1$
• D. ${}^{12}{C}_{10}$

Answer 1

Answer: (B), (C)

$12C_2{-}^5{C}_3{-}^4{C}_2{-}^3{C}_1$

Coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

= $(1-x^7)(1-x^8)(1-x^9)$$(1{+}^3{C}_{1}x{+}^4{C}_2{x}^2+........{+}^{12}{C}_{10}{x}^{10}+..........)$

Ignoring the powers higher than $x^{10}$

= $(1-x^7-x^8+x^{15})(1-x^9)$$(1{+}^3{C}_{1}x{+}^4{C}_2{x}^2+...........{+}^{12}{C}_{10}{x}^{10}+.............)$

= $(1-x^7-x^8)(1-x^9)$$(1{+}^3{C}_{1}x{+}^4{C}_2{x}^2+..........{+}^{12}{C}_{10}{x}^{10}+..............)$

Coefficient of $x^{10}$ in $(1-x^7-x^8-x^9)$$(1{+}^3{C}_{1}x{+}^4{C}_2{x}^2+............{+}^{12}{C}_{10}{x}^{10}+................)$

${}^{12}{C}_2{-}^5{C}_3{-}^4{C}_2{-}^3{C}_1{=}^{12}{C}_2{-}^5{C}_2{-}^4{C}_2{-}^3{C}_2$