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# Counting Question

0
23
1

Find the constant term in
$$( 2z - \frac{1}{z \sqrt{z}})^5$$

Feb 27, 2023

#1
0

To find the constant term in the expansion of $(2z - \frac{1}{z\sqrt{z}})^5$, we need to find the term that does not contain any powers of $z$.

The constant term can be obtained by selecting terms that contribute a power of $z^0$, which means we need to select one term from each of the two factors in the expansion. We can choose the $2z$ term or the $-\frac{1}{z\sqrt{z}}$ term from each factor.

For the $2z$ term, we need to choose it 0 times, 1 time, 2 times, 3 times, 4 times, or 5 times. Similarly, for the $-\frac{1}{z\sqrt{z}}$ term, we need to choose it 0 times, 1 time, 2 times, 3 times, 4 times, or 5 times.

We want to choose terms that result in a power of $z^0$, so we need to choose the $-\frac{1}{z\sqrt{z}}$ term 5 times, and the $2z$ term 0 times. Therefore, the constant term in the expansion is 60.

Feb 27, 2023