Let P be a product given by P = (x + a 1 ) (x + a 2 ) ......... (x + a n )
and Let S 1 = a 1 + a 2 + ....... + a n = 
, S 2 = 
S 3 = 
and so on,
then it can be shown that
P = x n + S 1 x n – 1 + S 2 x n – 2 + ......... + S n .
(i)The coefficient of x 8 in the expression (2 + x) 2 (3 + x) 3 (4 + x) 4 must be
Text Solution
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id),ii(c),iii(c)
(i) The expression (2 + x) 2 (3 + x) 3 (4 + x) 4 = (x + 2)(x + 2) (x + 3)(x + 3)(x + 3)(x + 4)(x + 4)(x + 4)(x + 4)
= x 9 + (2 + 2 + 3 + 3 + 3 + 4 + 4 + 4 + 4) x 8 + ..............
⇒ Co-efficient of x 8 = 29
(ii)Expression = x. x 2 . x 3 ...........x 20 
Let E = 
Now Co-efficient of x 203 in original expression
⇒ Co-efficient of x –7 in E.
But
E = 1 – 
+ 
– 
= Co-efficient of x –7 = –7 + 6 + 10 + 12 – 8 = 13
(iii)The Co-efficient of x 98 = (1.2 + 2.3 + ...........99.100)
= Sum of product of first 100 natural numbers taken two at a time
= 
[(1 + 2 + 3 + .......+ 100 2 ) 2 – (1 2 + 2 2 + 3 2 +..........+ 100 2 )]
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