For the expression \((1-x)^{100}\). Then sum of coefficient of first 50 terms is:
- \(^{99}C_{49}\)
- \(-\frac{^{100}C_{50}}{2}\)
- \(-^{99}C_{49}\)
- \(-^{101}C_{50}\)
The Correct Option is B
Solution and Explanation
Sum of coefficient of first 50 terms
\((t)\) \(= ^{100}C_0-^{100}C_1+...+^{100}C_{40}\)
Now
\(^{100}C_0-^{100}C_1+...+^{100}C_{100}=0\)
\(2[^{100}C_0-^{100}C_1+...]+ ^{100}C_{50}=0\)
\(\therefore \; t= -\frac{1}{2}^{100}C_{50}\)
The correct option is (B): \(-\frac{^{100}C_{50}}{2}\)
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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 .
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