Sagot :
Answer:
There are 1287 ways we can form a group of eight persons in a committee from 13 population.
Step-by-step explanation:
This is one of the applications in combinatorics. The concept of counting the number of distinct objects from a set of populations without regarding the order is what we called combination.
Some Combinatorics Topics
- Fundamental Concept of Counting
- Combination
- Permutation
Fundamental Concept of Counting states that if two or more events have a respective possible outcome, then taking all the events together, the product of the individual events is the total possible outcomes provided that all the events happen altogether.
Combination is the process of picking a set of objects from a population without regarding the order.
Permutation is the process of picking a set of objects from a population but this time, the order matters.
Solutions
The total population is 13. This is the sum of men and women ( nine men and four women ).
We are supposed to form a committee of eight people. As you observe, we don't really care who can be picked first or last, our concern is as long as we can form eight people. So basically, we disregard the order of the people being picked.
So, the appropriate counting technique to be used is the combination.
The formula for combination is given by [tex]\begin{aligned}_{n}C_r&=\frac{n!}{r!(n-r)!}\end{aligned}[/tex] where [tex]n[/tex] is the total population and [tex]r[/tex] is the number of distinct objects chosen from the population.
Substitute the value of [tex]n=13[/tex] and [tex]r=8[/tex] into the formula, then solve the right side of the equation.
[tex]\begin{aligned}_{n}C_r&=\frac{n!}{r!(n-r)!}\\_{13}C_8&=\frac{13!}{8!(13-8)!}\\&=\frac{13!}{8!5!}\\&=\frac{13\cdot 12 \cdot 11\cdot 10 \cdot 9\cdot 8!}{8!5!}\\&=\frac{13\cdot 12\cdot 11\cdot 10 \cdot 9}{5\cdot 4 \cdot 3\cdot 2 \cdot 1}\\&=\frac{154,440}{120}\\&=\boxed{1287}\end{aligned}[/tex]
This means to say that there are 1287 ways we can form a group of eight persons in a committee from 13 population.
To learn more about counting techniques, visit the following websites
- Circular Permutation: https://brainly.ph/question/493766
- Repeating objects in a Permutation: https://brainly.ph/question/2547472
- Counting Techniques: https://brainly.ph/question/2093332
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