The mean of coefficients of x, x2, ....., x7 in the binomial expansion of (2 + x)9 is?
The mean of coefficients of x, x2, ....., x7 in the binomial expansion of (2 + x)9 is?
Solution and Explanation
Sum of coefficients = \(^9C_1.2^8+^9C_2.2^7.....+^9C_7.2^2\)
\(\Rightarrow 3^9-2^9-19=19683-572-19 = 19152\)
Mean = \(\frac{sum}{7}=\frac{19152}{7}=2736\)
\(\therefore\) the correct answer is \(2736\).
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.