Answer:
The coefficient is 2520
Step-by-step explanation:
Given
[tex](x + y + z)^{10[/tex]
Required
The coefficient of [tex]x^3y^2z^5[/tex]
To do this, we make use of:
[tex]k = \frac{n!}{n_1!n_2!....n_m!}[/tex]
Where
[tex]k \to[/tex] coefficient
[tex]n = 10[/tex] the exponent of the given expression
[tex]n_1=3[/tex] -- exponent of x
[tex]n_2=2[/tex] --- exponent of y
[tex]n_3=5[/tex] --- exponent of z
So, we have:
[tex]k = \frac{n!}{n_1!n_2!....n_m!}[/tex]
[tex]k = \frac{10!}{3!2!5!}[/tex]
Expand
[tex]k = \frac{10*9*8*7*6*5!}{3!2!5!}[/tex]
[tex]k = \frac{10*9*8*7*6}{3!2!}[/tex]
[tex]k = \frac{30240}{3!2!}[/tex]
Expand the denominator
[tex]k = \frac{30240}{3*2*1*2*1}[/tex]
[tex]k = \frac{30240}{12}[/tex]
[tex]k = 2520[/tex]
