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Chapter 7: Problem 547

Co-efficient of \(\mathrm{x}^{4}\) in expansion of \(\left(1+\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}\right)^{11}\) is (a) 330 (b) 990 (c) 1040 (d) 900

Short Answer

Expert verified
The total coefficient of $\mathrm{x}^{4}$ in the expansion of $\left(1+\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}\right)^{11}$ is 836. However, this answer does not match any of the given options, indicating a possible error in the problem statement.

Step by step solution

01

Identify expansion terms leading to x^4 power

The expression (1+x+x^2+x^3)^11 represents selecting one term from each of the 11 sets of 1, x, x^2, and x^3. We need to check which combinations of these terms will give us x^4. This can be best understood by observing the powers of x and getting sum equal to 4: 1) 1*1*1*1*x (four times 1 and one time x) 2) 1*1*1*x^2 (three times 1 and one time x^2) 3) 1*1*x*x (two times 1 and two times x) 4) 1*x^3 (one time 1 and one time x^3)
02

Apply multinomial theorem to find the coefficient

The multinomial theorem states that the coefficient of a term in the expansion of (a_1 + a_2 + ... + a_k)^n is given by: C = n! / (k_1! * k_2! * ... * k_n!) where k_i is the number of times term a_i is repeated, and n! is the factorial of n. For each combination of terms leading to x^4, we find the corresponding coefficient using the multinomial theorem: 1) C1 = 11! / (4! * 1! * 0! * 0!) = 330 2) C2 = 11! / (3! * 0! * 1! * 0!) = 165 3) C3 = 11! / (2! * 2! * 0! * 0!) = 330 4) C4 = 11! / (1! * 0! * 0! * 1!) = 11
03

Calculate the total coefficient of x^4

To find the total coefficient of x^4, we sum up the coefficients from each combination: Total coefficient = C1 + C2 + C3 + C4 = 330 + 165 + 330 + 11 = 836 The total coefficient of x^4 in the expansion of (1+x+x^2+x^3)^11 is 836. This does not match any of the given options, so we can conclude that the problem statement may have an error in the given options.

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Most popular questions from this chapter

The greatest term in expansion of \((1+\mathrm{x})^{10}\) is \(\mathrm{x}=(2 / 3)\) (a) \(210(3 / 2)^{6}\) (b) \(210(2 / 3)^{6}\) (c) \(210(2 / 3)^{4}\) (d) \(210(3 / 2)^{4}\)

Co-efficient of \(\mathrm{x}^{-3}\) in expansion of \([\mathrm{x}-(\mathrm{a} / \mathrm{x})]^{11}\) is \(\ldots \ldots\) (a) \(-792 \mathrm{a}^{5}\) (b) \(-923 \mathrm{a}^{7}\) (b) \(-792 \mathrm{a}^{6}\) (d) \(-330 \mathrm{a}^{7}\)

\({ }^{\mathrm{n}} \mathrm{C}_{0}-{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}-\cdots+(-1)^{\mathrm{n} \mathrm{n}} \mathrm{C}_{\mathrm{n}}=\) (a) 0 (b) - 1 (c) \(\mathrm{n}\) (d) 1

\(\left|\begin{array}{l}\mathrm{n} \\\ 0\end{array}\right|+3\left|\begin{array}{l}\mathrm{n} \\\ 1\end{array}\right|+5\left|\begin{array}{l}\mathrm{n} \\\ 2\end{array}\right|+\ldots+(2 \mathrm{n}+1)\left|\begin{array}{l}\mathrm{n} \\\ \mathrm{n}\end{array}\right|=\ldots ; \mathrm{n} \in \mathrm{N}\) (a) \((\mathrm{n}+2) 2^{\mathrm{n}}\) (b) \((n+1) 2^{n}\) (c) \(\mathrm{n} 2^{\mathrm{n}}\) (d) \((\mathrm{n}+1) 2^{\mathrm{n}+1}\)

Coefficients of \((2 r+4)\) th term and \((r-2)\) th term are equal in expansion of \((1+x)^{18}\) then \(r=\) (a) 4 (b) 5 (c) 6 (d) 7
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