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Chapter 1: Problem 16

a) Find the coefficient of \(x^{2} y z^{2}\) in the expansion of \([(x / 2)+y-3 z]^{5}\). b) How many distinct terms are there in the complete expansion of $$ \left(\frac{x}{2}+y-3 z\right)^{5} ? $$ c) What is the sum of all coefficients in the complete expansion?

Short Answer

Expert verified
a) The coefficient of \(x^{2} y z^{2}\) is -10. b) The complete expansion consists of 21 distinct terms. c) The sum of all coefficients in the expansion is -243.

Step by step solution

01

Identify the specific term coefficient

To find the coefficient of \(x^{2} y z^{2}\) in the expansion of \((x/2+y-3z)^{5}\), first note that the power of x in the required term is 2, and in the algebraic expression the power of x is 1. Hence the number of times it occurs is 2. Similarly, the power of y is 1 in both the required term and the expression, while the power of z is 2 in the required term and 1 in the expression, hence it occurs twice. So, following the binomial theorem \((x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k}\), the coefficient of \(x^{2} y z^{2}\) is \(\binom{5}{2, 1, 2}((1/2)^{2})((-3)^{2})\). Calculating, the coefficient is -10
02

Compute the distinct terms

The number of distinct terms in the expansion of \((a+b+c)^{n}\) is given by \(\binom{n+3-1}{n} = \binom{5+3-1}{5} = \binom{7}{5} = 21\). Therefore, there are 21 distinct terms in the expansion.
03

Sum of all coefficients

To find the sum of all coefficients in the expansion, substitute all the variables in the expression with 1. This is because by substituting any variable with 1, the coefficient remains the same while the variable itself does not contribute to the sum. Hence, substituting \(x = y = z = 1\) in \((x/2+y-3z)^{5}\) results into \((1/2+1-3)^{5} = (-3/2)^{5} = -243\

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