Question:
If the sum of the coefficients in the expansion of $(a + b)^n$ is 4096, then the greatest coefficient in the expansion is
If the sum of the coefficients in the expansion of $(a + b)^n$ is 4096, then the greatest coefficient in the expansion is
Updated On: Jul 5, 2022
- 1594
- 792
- 924
- 2924
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The Correct Option is C
Solution and Explanation
We have $2^n = 4096 = 2^{12 } \, \Rightarrow \, n = 12;$
the greatest coeff = coeff of middle term.
So middle term
$ = t_7 , ; t_7 = t_{6+1}$
$\Rightarrow \,$ coeff of $ t_7 = {^{12}C_{6}} = \frac{12!}{6! 6!} = 924 $.
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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.