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Question

If a, b, c are positive integers, then the co-efficient of x in $capital delta equals open vertical bar table row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of a end exponent end cell cell open parentheses 1 plus x close parentheses to the power of b c end exponent end cell row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of b end exponent end cell cell open parentheses 1 plus x close parentheses to the power of c a end exponent end cell row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of c end exponent end cell cell open parentheses 1 plus x close parentheses to the power of a b end exponent end cell end table close vertical bar$ is

a + b + c
abc
a² + b² + c²
0.

The correct answer is: a² + b² + c²

Each term in the expansion of ∆ is of the form $open parentheses 1 plus x close parentheses to the power of m end exponent$ , where m is a +ve integer.
We have $open parentheses 1 plus x close parentheses to the power of m end exponent$ = $1 plus m x plus fraction numerator 1 over denominator 2 end fraction m open parentheses m minus 1 close parentheses x to the power of 2 end exponent plus......$
Co-eff. of x in $open parentheses 1 plus x close parentheses to the power of m end exponent$ = ∆’ (0) = 0
We have
$capital delta ´ open parentheses x close parentheses equals open vertical bar table row 1 cell open parentheses 1 plus x close parentheses to the power of a end exponent end cell cell open parentheses 1 plus x close parentheses to the power of b c end exponent end cell row 1 cell open parentheses 1 plus x close parentheses to the power of b end exponent end cell cell open parentheses 1 plus x close parentheses to the power of c a end exponent end cell row 1 cell open parentheses 1 plus x close parentheses to the power of c end exponent end cell cell open parentheses 1 plus x close parentheses to the power of a b end exponent end cell end table close vertical bar plus open vertical bar table row cell 1 plus x end cell cell a open parentheses 1 plus x close parentheses to the power of a minus 1 end exponent end cell cell open parentheses 1 plus x close parentheses to the power of b c end exponent end cell row cell 1 plus x end cell cell b open parentheses 1 plus x close parentheses to the power of b minus 1 end exponent end cell cell open parentheses 1 plus x close parentheses to the power of c a end exponent end cell row cell 1 plus x end cell cell c open parentheses 1 plus x close parentheses to the power of c minus 1 end exponent end cell cell open parentheses 1 plus x close parentheses to the power of a b end exponent end cell end table close vertical bar plus open vertical bar table row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of a end exponent end cell cell b c open parentheses 1 plus x close parentheses to the power of b c minus 1 end exponent end cell row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of b end exponent end cell cell c a open parentheses 1 plus x close parentheses to the power of c a minus 1 end exponent end cell row cell 1 plus x end cell cell open parentheses 1 plus x close parentheses to the power of c end exponent end cell cell a b open parentheses 1 plus x close parentheses to the power of a b minus 1 end exponent end cell end table close vertical bar$
$therefore capital delta ´ open parentheses 0 close parentheses equals open vertical bar table row 1 1 1 row 1 1 1 row 1 1 1 end table close vertical bar plus open vertical bar table row 1 a 1 row 1 b 1 row 1 a 1 end table close vertical bar plus open vertical bar table row 1 1 cell b c end cell row 1 1 cell c a end cell row 1 1 cell a b end cell end table close vertical bar equals 0 plus 0 plus 0 equals 0$

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