Answered

From a bag containing nine white feathers and two black feathers, you draw three feathers at random. Which of these could be used to represent the number of ways of selecting three feathers from 11?

A. [tex]${ }_3 C_{11}$[/tex]
B. [tex]${ }_1 C_{11}$[/tex]
C. [tex]${ }_1 C_3$[/tex]
D. [tex]${ }_{11} C_3$[/tex]



Answer :

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]