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Chapter 7: Problem 498

In the expansion of \((\mathrm{x}+\mathrm{y})^{13}\) the co-efficient of \(3 \mathrm{rd}\) term and th terms are equal. (a) 12 (b) 11 (c) 8 (d) 13

Short Answer

Expert verified
None of the given options match the obtained value of 'n', which is 5. There might be a mistake in the given options, or the given data in the question might require a revision.

Step by step solution

01

Find the formula for coefficients using the binomial theorem

According to the binomial theorem, the general term in the expansion of \((x+y)^n\) is given by: \[T_r = \binom{n}{r-1}x^{n-r+1}y^{r-1}\] where \(T_r\) represents the \(r^{th}\) term in the expansion.
02

Find the 3rd and 4th terms in the expansion

Given that the coefficients of the 3rd and 4th terms are equal, we need to find the coefficients of these terms using the binomial theorem. For the 3rd term, \(r=3\): \[T_3 = \binom{n}{2}x^{n-1}y^2\] For the 4th term, \(r=4\): \[T_4 = \binom{n}{3}x^{n-2}y^3\]
03

Equate the coefficients of the 3rd and 4th terms

Since the coefficients of the 3rd and 4th terms are equal, we need to equate the coefficients and solve for 'n': \[\binom{n}{2} = \binom{n}{3}\]
04

Solve for 'n'

Using the combination formula, we can rewrite the equation as: \[\frac{n!}{2!(n-2)!} = \frac{n!}{3!(n-3)!}\] Now, we can simplify and solve for 'n': \[\frac{1}{2(n-2)} = \frac{1}{3(n-3)}\] Cross-multiplying, we get: \[3(n-3) = 2(n-2)\] Expanding and solving for 'n', we get: \[3n - 9 = 2n - 4\] \[n = 5\] Given the options (a) 12, (b) 11, (c) 8, and (d) 13, none of them match with the obtained value of 'n'. There might be a mistake in the given options, or the given data in the question might require a revision.

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Most popular questions from this chapter

Let \(\mathrm{x}>-1\) then statement \((1+\mathrm{x})^{\mathrm{n}}>1+\mathrm{n} \mathrm{x}\) is true for (a) \(\forall \mathrm{n} \in \mathrm{N}\) (b) \(\forall n>1\) (c) \(\forall \mathrm{n}>1\) and \(\mathrm{x} \neq 0\) (d) \(\forall \mathrm{n} \in \mathrm{R}\)

Remainder when \(2^{2000}\) is divided by 17 is......... (a) 1 (b) 2 (c) 8 (d) 12

\(\mathrm{s}(\mathrm{k}): 1+3+5+\ldots+(2 \mathrm{k}-1)=3+\mathrm{k}^{2}\) then which statement is true ? (a) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (b) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (c) s (1) is true (d) Result is proved by Principle of Mathematical induction

Coefficients of middle terms in expansion of \(\left[2-\left(\mathrm{x}^{3} / 3\right)\right]^{7}\) are... (a) \(-[(560) /(27)],-[(280) /(81)]\) (b) \([(560) /(27)],-[(280) /(81)]\) (c) \(-[(560) /(27)],[(280) /(81)]\) (d) \([(560) /(27)],[(280) /(81)]\)

Number of irrational terms in expansion of \(\left[4^{(1 / 5)}+7^{(1 / 10)}\right]^{45}=\) (a) 40 (b) 5 (c) 41 (d) 8
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