Question

The coefficient of $x^{10}$ in the expansion of $(1+x{)}^2(1+x^2{)}^3(1+x^3{)}^4$ is equal to :

• A. $44$
• B. $50$
• C. $52$
• D. $56$

Answer 1

The correct option is C $52$

The expansions are:
$(1+x{)}^2=1+2x+x^2...(1)$
$(1+x^2{)}^3=1+3x^2+3x^4+x^6....(2)$
$(1+x^3{)}^4=1+4x^3+6x^6+4x^9+x^{12}...(3)$
The possible combinations for $x^{10}=(x^0,x^4,x^6);(x^1,x^0,x^9);(x^1,x^6,x^3);(x^2,x^2,x^6)$
Case1:
If combination is $x^0,x^4,x^6$
The coefficient is $1\times 3\times 6=18$
Case 2:
If combination is $x^1,x^0,x^9$
The coefficient is $2\times 1\times 4=8$
Case 3:
If combination is $x^1,x^6,x^3$
The coefficient is $2\times 1\times 4=8$
Case 4:
If combination is $x^2,x^2,x^6$
The coefficient is $1\times 3\times 6=18$
Hence, The coefficient of $x^{10}=18+8+8+18=52$