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 \(2736\).
Top Questions on Binary operations
The number of rational terms in the expansion of (33/4 + 53/2)60?
- JEE Main - 2023
- Mathematics
- Binary operations
- Let ⊙ be a binary operation on Q - {0} defined by a⊙b=\(\frac{a}{b}\). Then 1⊙(2⊙(3⊙4)) is equal to
- KEAM - 2022
- Mathematics
- Binary operations
- For any two positive rational numbers m and n, a binary operation * is defined by \(m*n=\frac{m+n}{3}\), then \(\frac{7}{2}*\frac{5}{2}\) is equal to
- KEAM - 2021
- Mathematics
- Binary operations
- If A = {a, b, c}, then the number of binary operations on A is
- KCET - 2020
- Mathematics
- Binary operations
- If $A = \{a,b,c\}$, then the number of binary operations on $A$ is
- KCET - 2020
- Mathematics
- Binary operations
Questions Asked in JEE Main exam
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Relations and functions
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the river bank is _________ cm. (Take \( g = 10 \, {m/s}^2 \)).
- JEE Main - 2025
- System of Particles & Rotational Motion
- 'X' is the number of acidic oxides among $ VO_2, V_2O_3, CrO_3, V_2O_5 $ and $ Mn_2O_7 $. The primary valency of cobalt in $ [Co(H_2NCH_2CH_2NH_2)_3]_2 (SO_4)_3 $ is Y. The value of X + Y is :
- JEE Main - 2025
- Inorganic chemistry
- The kinetic energy of translation of the molecules in 50 g of CO\(_2\) gas at 17°C is:
- JEE Main - 2025
- The Kinetic Theory of Gases
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.