If the sum of the co-efficients of all the positive even powers of x in the binomial expansion of \((2x^3+\frac{3}{x})^{10} \) is 510 – β·39, the β is equal to ____.
Correct Answer: 83
Solution and Explanation
The correct answer is: 83.
\(T_{r+1}=10C_r(2x^3)^{10-r}(\frac{3}{x})^r\)
\(=C_r^{10}2^{10-r}3^rx^{30-4r}\)
So, r ≠ 8, 9, 10
Sum of required Coeff.
\((2.1^3+\frac{3}{1})^{10}(c^{10}_82^23^8+c^{10}_92^13^9+c^{10}_{10}2^03^{10})\)
\(β=\frac{4}{3}.^{10}c_8+20+3=83\)
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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 .