ISI2015-DCG-6

Question

The coefficient of $x^2$ in the product $(1+x)(1+2x)(1+3x) \dots (1+10x)$ is

• A. $1320$
• B. $1420$
• C. $1120$
• D. None of these

Answer 1

ignore multiplications that give $x^i$ where $i>2$

$(1+x)(1+2x)(1+3x)(1+4x)(1+5x)(1+6x)(1+7x)(1+8x)(1+9x)(1+10x)$

$=(1+3x+2x^2)(1+7x+12x^2)(1+11x+30x^2)(1+15x+56x^2)(1+19x+90x^2)$

$=(1+7x+12x^2+3x+21x^2+2x^2)(1+15x+56x^2+11x+165x^2+30x^2)(1+19x+90x^2)$

$=(1+10x+35x^2)(1+26x+251x^2)(1+19x+90x^2)$

$=(1+26x+251x^2 + 10x + 260x^2 +35x^2)(1+19x+90x^2)$

$=(1+ 36x+546x^2  )(1+19x+90x^2)$

$=(1+19x+90x^2+36x+684x^2 +546x^2)$

So the coeffecient of $x^2 = 90+684+546= 1320$

$\therefore$ Option $A.$ is correct..