Question
Question: The coefficient of \(x^{5}\) in the expansion of \((x^{2} - x - 2)^{5}\) is...
The coefficient of x5 in the expansion of (x2−x−2)5 is
A
– 83
B
– 82
C
– 81
D
0
Answer
– 81
Explanation
Solution
Coefficient of x5 in the expansion of (x2−x−2)5 is
∑n1!.n2!n3!5!(1)n1(−1)n2(−2)n3.
where n1+n2+n3=5 and n2+2n3=5. The possible value of n1,n2 and n3 are shown in margin
n1 n2 n3
1 3 1
2 1 2
0 5 0
∴ The coefficient of x5
= 1!3!1!5!(1)1(−1)3(−2)1+2!1!2!5!(1)2(−1)1(−2)2 +
0!5!0!5!(1)0(−1)5(−2)0 = 40−120−1=−81