To solve the given problem, let's first identify what it is asking us to do. We need to determine which notation correctly represents the number of ways to select 3 feathers out of 11 total feathers.
Here's the step-by-step explanation:
1. Understanding the Problem:
- We have a total of 11 feathers: 9 white feathers and 2 black feathers.
- We need to find the number of ways to select 3 feathers in any possible combination out of these 11 feathers.
2. Choosing 3 out of 11:
- The problem involves calculating the number of combinations, which is a standard combinatorics problem.
- The combination formula used to determine the number of ways to choose k items from n items is usually written as "n choose k" and is mathematically represented as [tex]\( C(n, k) \)[/tex] or [tex]\( \binom{n}{k} \)[/tex].
3. Evaluating the Given Options:
- Let’s go through each of the options to see which one correctly represents choosing 3 feathers from a total of 11 feathers:
- Option 1: [tex]\({ }_3 C_{11}\)[/tex]
- This notation would represent choosing 11 feathers out of 3, which doesn't make sense because you cannot choose more items than you have. It should be the other way around: choosing some out of 11.
- Option 2: [tex]\({ }_1 C_{11}\)[/tex]
- This notation represents choosing 11 feathers out of 1, which again doesn't make sense for the same reason as above.
- Option 3: [tex]\({ }_1 C_3\)[/tex]
- This notation represents choosing 3 feathers out of 1, which is not what the problem is asking.
- Option 4: [tex]\({ }_{11} C_3\)[/tex]
- This notation correctly represents choosing 3 feathers from a total of 11 feathers, exactly as required by the problem.
To conclude, the correct notation to represent the number of ways of selecting three feathers from a total of 11 feathers is:
[tex]\[ \boxed{ { }_{11} C_3 } \][/tex]
