If the coefficient of xⁿ in (1 + x)^l0l (1 - x + x2)^100 is nonzero,
then n cannot be of the form
(1) 3r + 1
(2) 3r
(3) 3r + 2
(4) none of these
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Answer:
This can be re-written as
(1+x)[(1+x)(1−x+x
2
)]
100
=(1+x)(1+x
3
)
100
=(1+x
3
)
100
+x(1+x
3
)
100
Hence coefficient of x
n
will be
=
100
C
3
n
+
100
C
3
n−1
...(i)
Hence from i, we can say the n can of type , 3r or 3r+1 but not of type 3r+2.
Where rϵW
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