The coefficient of x28 in the expansion of 1-x3-x630 is A- 30 C 6B- 30 C 3- 1D- 0

The correct option is D 0(1+x^3 x^6)^30 nbsp;= ∑_k_1+k_2+k_3 = 3030!k_1!k_2!k_3! (1)^k_1(x^3)^k_2( x^6)^k_3 nbsp;= nbsp;∑_k_1+k_2+k_3 = 3030!k_1!k_2!k_ ...

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Standard XII

Mathematics

Multinomial Expansion

The coefficie...

Question

The coefficient of $x^{28}$ in the expansion of $(1 + x^{3} - x^{6})^{30}$ is

1

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^{30} C_{6}

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^{30} C_{3}

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Solution

The correct option is B $0$
$(1 + x^{3} - x^{6})^{30} = \sum k_{1} + k_{2} + k_{3} = 30 \frac{30!}{k_{1}! k_{2}! k_{3}!} (1)^{k_{1}} (x^{3})^{k_{2}} (- x^{6})^{k_{3}} = \sum k_{1} + k_{2} + k_{3} = 30 \frac{30!}{k_{1}! k_{2}! k_{3}!} (- 1)^{k_{3}} x^{3 k_{2} + 6 k_{3}}$

Now, for coefficient of $x^{28}$
$3 k_{2} + 6 k_{3} = 28 \Rightarrow k_{2} + 2 k_{3} = \frac{28}{3}$
As $k_{2}, k_{3} \in W$ , so it is not possible
Therefore, coefficient of $x^{28}$ is $0$ .

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Multinomial Expansion

Standard XII Mathematics

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The coefficient of x28 in the expansion of (1+x3−x6)30 is

The correct option is B 0(1+x3−x6)30=∑k1+k2+k3=3030!k1!k2!k3!(1)k1(x3)k2(−x6)k3=∑k1+k2+k3=3030!k1!k2!k3!(−1)k3x3k2+6k3 Now, for coefficient of x28 3k2+6k3=28⇒k2+2k3=283 As k2,k3∈W, so it is not possible Therefore, coefficient of x28 is 0.

The coefficient of $x^{28}$ in the expansion of $(1 + x^{3} - x^{6})^{30}$ is