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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

\(\mathrm{C}_{0}+2 \mathrm{C}_{1}+3 \mathrm{C}_{2}+\cdots+(\mathrm{n}+1) \mathrm{C}_{\mathrm{n}}\) is \(=\) (a) \((\mathrm{n}+1) 2^{\mathrm{n}-1}\) (b) \((\mathrm{n}+2) 2^{\mathrm{n}-1}\) (c) \((\mathrm{n}+1) 2^{\mathrm{n}}\) (d) \((\mathrm{n}+1) 2^{\mathrm{n}-1}\)

Middle term in expansion of \([(2 / x)-3 x y]^{12}\) is (a) \(14370048 \mathrm{y}^{6}\) (b) \(14370024 \mathrm{y}^{6}\) (c) \(43110144 \mathrm{y}^{6}\) (d) \(43110124 \mathrm{y}^{6}\)

th term is constant term in expansion of \(\left[\left(3 / x^{2}\right)+(\sqrt{x} / 3)\right]^{10}, x \neq 0\) (a) 4 (b) 7 (c) 8 (d) 9

The Co-efficient of \(x^{3}\) in \(\left(1-x+x^{2}\right)^{5}\) is (a) \(-30\) (b) \(-20\) (c) \(-10\) (d) 30

The expansion of \(\left[\mathrm{x}+\sqrt{\left(\mathrm{x}^{3}-1\right)}\right]^{5}+\left[\mathrm{x}-\sqrt{\left(\mathrm{x}^{3}-1\right)}\right]^{5}\) is a polynomial of degree (a) 5 (b) 6 (c) 7 (d) 8
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