Answer:
There are 8218 possible ways that a committee of 6 can be chosen if it must consist of at least 4 executive board members.
Step-by-step explanation:
This is a problem that involves combination. Combination is a combinatoric concept that deals with the number of distinct objects chosen from a set of objects disregarding the order of the objects.
The formula for the combination is given by [tex]\begin{aligned}_nC_r=\frac{n!}{r!(n-r)!}\end{aligned}[/tex].
Some Fundamental Principle of Counting Concepts
Solution:
The phrase "at least 4 executive boards" means that the number of executive board member can be 4 or it can be greater than 4.
We can take the combination of each case, then add all the possible outcomes.
Case 1: 4 from executive board members and 2 from general members
We can choose 4 out of 8 and 2 out of 15 to form the committee.
[tex]\begin{aligned}_8C_4 \times _{15}C_2 &=\frac{8!}{4!(8-4)!}\times \frac{15!}{2!(15-2)!}\\&=70\times105\\&=7350\end{aligned}[/tex]
There are 7350 ways we can choose 4 from the executive board members and 2 from general members to form the committee of 6 members.
Case 2: 5 from executive board members and 1 from general members
We can choose 5 out of 8 and 1 out of 15 to form the committee.
[tex]\begin{aligned}_8C_5 \times _{15}C_1 &=\frac{8!}{5!(8-5)!}\times \frac{15!}{1!(15-1)!}\\&=56\times15\\&=840\end{aligned}[/tex]
There are 840 ways we can choose 5 from the executive board members and 1 from general members to form the committee of 6 members.
Case 3: 6 from executive board members.
We can choose 6 out of 8 to form the committee.
[tex]\begin{aligned}_8C_6&=\frac{8!}{6!(8-6)!}\\&=28 \end{aligned}[/tex]
There are 28 ways we can choose 6 from the executive board members to form the committee of 6 members.
Adding all the possible cases, we have 7350 + 840 + 28 = 8218
Thus, there are 8218 possible ways we can choose a committee of 6 consisting of at least 4 executive board members.
To learn more about combinations, go to
- Example of Combination and Permutation: https://brainly.ph/question/1994449
- Reflection about combination: https://brainly.ph/question/504738
- Difference between combination and permutation: https://brainly.ph/question/103634
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