Let's solve the given problem step-by-step.
The question gives us two pieces of information:
1. [tex]\( m = \binom{-2}{3} \)[/tex]
2. [tex]\( \omega = 13 \)[/tex]
### Part (a): Finding [tex]\( \text{ran} \)[/tex]
Since the question does not provide enough context or clarity about what [tex]\( \text{ran} \)[/tex] refers to, we need more information to proceed with this part. Therefore, without additional context, [tex]\( \text{ran} \)[/tex] cannot be determined from the given information.
### Part (b): Finding [tex]\( \omega \)[/tex]
Given directly in the problem:
[tex]\[ \omega = 13 \][/tex]
### Finding [tex]\( m \)[/tex]
The binomial coefficient [tex]\( \binom{-2}{3} \)[/tex] where [tex]\( n < k \)[/tex] is given and typically, the binomial coefficient [tex]\( \binom{n}{k} \)[/tex] is defined as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
However, it’s important to note that binomial coefficients where [tex]\( n \)[/tex] is less than [tex]\( k \)[/tex] are defined to be 0 because it's not possible to choose more elements than are available. Thus:
[tex]\[ m = \binom{-2}{3} = 0 \][/tex]
### Summary
- For [tex]\( m \)[/tex]: [tex]\( m = 0 \)[/tex]
- For [tex]\( \omega \)[/tex]: [tex]\( \omega = 13 \)[/tex]
So, the results are:
a) [tex]\( \text{ran} \)[/tex]: Not determinable with the given information.
b) [tex]\( \omega = 13 \)[/tex]
This completes the problem based on the provided information.
