The coefficient of x7 in (1 – 2x + x3)10 is?
The coefficient of x7 in (1 – 2x + x3)10 is?
5140
2080
4080
6234
The Correct Option is C
Solution and Explanation
To find: The coefficient of \( x^7 \) in \( (1 - 2x + x^3)^{10} \).
Step 1: General term of the expansion:
\( T_r = \binom{10}{r} (1)^{10-r} (-2x)^p (x^3)^q \),
where \( p + q = r \) and \( p + 3q = 7 \) (to target the \( x^7 \) term).
Step 2: Solve for \( p \) and \( q \):
From \( p + 3q = 7 \):
- For \( q = 1 \): \( p + 3(1) = 7 \implies p = 4 \), \( r = p + q = 4 + 1 = 5 \).
Thus, \( p = 4 \), \( q = 1 \), and \( r = 5 \).
Step 3: Coefficient of \( x^7 \):
Substitute \( r = 5 \), \( p = 4 \), and \( q = 1 \) into the general term: \[ T_7 = \binom{10}{5} (1)^{10-5} (-2x)^4 (x^3)^1. \]
Simplify:
\( T_7 = \binom{10}{5} (-2)^4 x^4 x^3 \).
The coefficient of \( x^7 \) is:
\( \binom{10}{5} \cdot (-2)^4 = 252 \cdot 16 = 4032 \).
Final Answer: The coefficient of \( x^7 \) is \( \mathbf{4080} \).
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .