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Multinomial Expansion

If the number...

Question

If the number of terms in the expansion of $(x + y + z)^{n}$ is $36$ , then the value of $n$ is

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Solution

Given expansion $(x + y + z)^{n}$

The number of terms is
$^{n + 3 - 1} C_{3 - 1} = 36 \Rightarrow^{n + 2} C_{2} = 36 \Rightarrow \frac{(n + 2) (n + 1)}{2} = 36 \Rightarrow n^{2} + 3 n + 2 = 72 \Rightarrow n^{2} + 3 n - 70 = 0 \Rightarrow (n + 10) (n - 7) = 0 \Rightarrow n = 7 (∵ n \in N)$

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