There are 16 people interested in helping with a committee. In how many ways could a committee be chosen, assuming the committee contains at
least one person.
review the power set
There are 2^16 - 1 non-empty subsets of a set of 16 elements
I don't understand it. I've done similar questions before but this one is different. Do you mind explaining it a bit more? Also what power set?
ever hear of google? search for "power set"
Suppose you have 3 elements. You can make all the non-empty subsets as follows:
{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
There are 2^3 - 1 = 7 such subsets
The nth row of Pascal's Triangle tells how many subsets of k elements there, with k going from 0 to n. The numbers on the nth row add up to 2^n.
To find the number of ways a committee can be chosen, we can use the principle of combinations. In this case, we need to find the number of ways to choose a committee from a group of 16 people.
Since the committee must contain at least one person, we can start by considering all possible committee sizes: 1 person, 2 people, 3 people, and so on, up to 16 people.
To determine the number of committees for each committee size, we can use binomial coefficients. The number of ways to choose "k" objects from a set of "n" objects is given by the formula:
n C k = n! / [(n-k)! * k!]
Using this formula, we can calculate the number of committees for each committee size and then sum them up to find the total number of ways a committee can be chosen. Let's calculate it step by step:
1. For a committee size of 1 person:
- There are 16 possible ways to choose 1 person.
2. For a committee size of 2 people:
- There are 16 possible ways to choose the first person.
- There are 15 possible ways to choose the second person.
- However, since the order of selection does not matter (e.g., AB is the same as BA), we need to divide by 2 to account for this overcounting.
- So, the number of ways to choose 2 people is (16 * 15) / 2.
3. For a committee size of 3 people:
- There are 16 possible ways to choose the first person.
- There are 15 possible ways to choose the second person.
- There are 14 possible ways to choose the third person.
- Since the order of selection does not matter, we need to divide by 3! (3 factorial).
- So, the number of ways to choose 3 people is (16 * 15 * 14) / (3 * 2 * 1).
4. Continue this process for committee sizes of 4, 5, ..., up to 16 people.
Finally, sum up the number of committees for each committee size to find the total number of ways a committee can be chosen:
Total number of ways = (1) + (16*15/2) + (16*15*14/(3*2*1)) + ... + (16!) / [(16-k)! * k!]
In this case, since we are considering committee sizes from 1 to 16, we need to sum up the formula for k = 1 to k = 16.
Calculating this sum may take some time manually, but using a calculator or spreadsheet software can significantly simplify the process.