If you happen to be viewing the article In how many ways can a committee of 7 students be chosen from 9 juniors and 9 seniors If there must be 4 sensors in the committee? ? on the website Math Hello Kitty, there are a couple of convenient ways for you to navigate through the content. You have the option to simply scroll down and leisurely read each section at your own pace. Alternatively, if you’re in a rush or looking for specific information, you can swiftly click on the table of contents provided. This will instantly direct you to the exact section that contains the information you need most urgently.
Contents
In how many ways can a committee of 7 students be chosen from 9 juniors and 9 seniors If there must be 4 sensors in the committee?
A. 1764
B. 5040
C. 10584
D. 12365
Answer : There are 10,584 ways to form the committee with 4 seniors and 3 juniors.
To form the committee, we need to select 4 seniors and 3 juniors from the respective pools. We’ll calculate this using combinations:
Number of ways to choose 4 seniors from 9 seniors: 9 choose 4 = 9! / (4! * (9-4)!) = 9! / (4! * 5!) = 126
Number of ways to choose 3 juniors from 9 juniors: 9 choose 3 = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = 84
To find the total number of ways to form the committee, we multiply the number of ways to choose seniors by the number of ways to choose juniors: Total ways = 126 * 84 = 10,584
Article continues below advertisement
So, there are 10,584 ways to form the committee with 4 seniors and 3 juniors.
Combinations and Permutations
Combinations and permutations are fundamental concepts in combinatorics, which is a branch of mathematics concerned with counting, arranging, and organizing objects. Here’s a brief explanation of each:
Article continues below advertisement
Article continues below advertisement
-
Permutations: Permutations refer to the arrangements of objects in a particular order. For a set of n distinct objects, the number of permutations of k objects taken from the set (where order matters and repetition is not allowed) is denoted by P(n, k) or sometimes as “n P k”, and it is calculated using the formula:
P(n, k) = n * (n-1) * (n-2) * … * (n-k+1)
This is also written in terms of factorials as:
P(n, k) = n! / (n-k)!
Example: If you have 5 different books and you want to arrange 3 of them on a shelf in a specific order, the number of permutations would be P(5, 3) = 5! / (5-3)! = 5! / 2! = 5 * 4 * 3 = 60.
-
Combinations: Combinations refer to the selections of objects from a group, where the order doesn’t matter. For a set of n distinct objects, the number of combinations of k objects taken from the set (where order doesn’t matter and repetition is not allowed) is denoted by C(n, k) or sometimes as “n C k”, and it is calculated using the formula:
C(n, k) = n! / (k! * (n-k)!)
Example: If you have 5 different books and you want to select 3 of them to take home without caring about the order, the number of combinations would be C(5, 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3) / (3 * 2) = 10.
In summary, permutations deal with arrangements where order matters, while combinations deal with selections where order doesn’t matter.
Thank you so much for taking the time to read the article titled In how many ways can a committee of 7 students be chosen from 9 juniors and 9 seniors If there must be 4 sensors in the committee? written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!
Source: Math Hello Kitty
Categories: Math
