The coefficient of x^n-2 in the polynomial (x-1)(x-2) ..... (x − n) is
a) - 1/24 n(n+1)(n-1)(3n+2)
b) 1/24 n(n^2- 1)(3n+2)
c) n(n +1)(2n+1)/6
d)n(n+1)(2n+1)/24
PLEASE ANSWER THIS QUESTION CORRECTLY
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Answer:
a) is the correct answer,
There are total of n brackets. The term x
n−2
will be formed when integers are chosen from any two brackets and x is chosen from all the other brackets and multiplied.
Thus, the Coefficient of x
n−2
is
C=(1×2+1×3+...+1×n)+(2×3+2×4+..+2×n)+...+((n−1)×n)
=(
2
(n)(n+1)
−1)+2(
2
(n)(n+1)
−1−2)+...+(n−1)(
2
(n)(n+1)
−(1+2+3+..+(n−1)))
={(1+2+3+...+(n−1))(
2
(n)(n+1)
)}−{1+2(1+2)+3(1+2+3)+...+(n−1)(1+...+n−1)}
={(
2
(n−1)(n)
)(
2
(n)(n+1)
)}−{∑
1
n−1
k(
2
(k)(k+1)
)}
={
4
n
2
(n
2
−1)
}−{∑
1
n−1
2
k
3
+k
2
}
={
4
n
2
(n
2
−1)
}−
2
1
{(
2
(n−1)(n)
)
2
+
6
(n−1)n(2n−1)
}
=
4
n(n−1)
{n(n+1)−
2
n(n−1)
−
3
2n−1
}
=
4
n(n−1)
{
2
n(n+3)
−
3
2n−1
}
=
4
n(n−1)
{
6
3n
2
+9n−4n+2
}
=
4
n(n−1)
{
6
(3n+2)(n+1)
}
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