In the expansion of (1 + x + y + z)^4,the rat | vedclass

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Question

(A) The general term in the multinomial expansion of $(1 + x + y + z)^4$ is given by $\frac{4!}{n_1! n_2! n_3! n_4!} (1)^{n_1} (x)^{n_2} (y)^{n_3} (z)

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$1 : 1 : 2$

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(A) The general term in the multinomial expansion of $(1 + x + y + z)^4$ is given by $\frac{4!}{n_1! n_2! n_3! n_4!} (1)^{n_1} (x)^{n_2} (y)^{n_3} (z)^{n_4}$,where $n_1 + n_2 + n_3 + n_4 = 4$.$1$. For the coefficient of $x^2y$,we have $n_2=2, n_3=1, n_4=0$,so $n_1=1$. The coefficient is $\frac{4!}{1! 2! 1! 0!} = \frac{24}{2} = 12$.$2$. For the coefficient of $xy^2z$,we have $n_2=1, n_3=2, n_4=1$,so $n_1=0$. The coefficient is $\frac{4!}{0! 1! 2! 1!} = \frac{24}{2} = 12$.$3$. For the coefficient of $xyz$,we have $n_2=1, n_3=1, n_4=1$,so $n_1=1$. The coefficient is $\frac{4!}{1! 1! 1! 1!} = 24$.The ratio is $12 : 12 : 24$,which simplifies to $1 : 1 : 2$.

In the expansion of $(1 + x + y + z)^4$,the ratio of the coefficients of $x^2y$,$xy^2z$,and $xyz$ is:

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