First slide

Binomial theorem for positive integral Index

Question

$Coefficient of x^{6} in (1 + x) {(1 + x^{2})}^{2} {(1 + x^{3})}^{3} ........ {(1 + x^{n})}^{n}, n \geq 6, is$

Moderate

A

26
B

28
C

30
D

35

Solution

The coefficient of $x^{6}$ in the given expression
= Coefficient of $x^{6}$ in $(1 +^{6} C_{1} x^{6}) (1 +^{5} C_{1} x^{5}) (1 +^{4} C_{1} x^{4}) (1 +^{3} C_{1} x^{3} +^{3} C_{2} x^{6}) (1 +^{2} C_{1} x^{2} +^{2} C_{2} x^{4}) (1 + x)$
$= Coefficient of x^{6} in (1 + 6 x^{6} + 5 x^{5} + 4 x^{4}) (1 + 2 x^{2} + 3 x^{3} + x^{4} + 6 x^{5} + 3 x^{6}) (1 + x)$
$= Coefficient of x^{6} in (11 x^{5} + 17 x^{6}) (1 + x) = 28$

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