1.There are 20 people in a club. In how many ways can the club elect a president, a secretary and a treasurer, on person can hold more than one office?
2. How many ways can a committee of 4 people be selected from a group of 10 people?
20P3
10C4
1. To find the number of ways the club can elect a president, secretary, and treasurer, we can use the concept of permutations. Since one person can hold more than one office, the order in which the positions are filled matters.
To calculate the number of ways to elect the president, there are 20 candidates to choose from. Therefore, there are 20 possible choices for the president.
Similarly, for the position of secretary, once the president has been chosen, there are now 19 remaining candidates to choose from since one person can hold multiple positions. Therefore, there are 19 possible choices for the secretary.
For the position of treasurer, once the president and secretary have been chosen, there are now 18 remaining candidates to choose from. Therefore, there are 18 possible choices for the treasurer.
To find the total number of ways, we multiply the number of choices for each position:
Total number of ways = 20 * 19 * 18 = 6840 ways.
Therefore, there are 6840 ways in which the club can elect a president, a secretary, and a treasurer, allowing one person to hold multiple offices.
2. To find the number of ways a committee of 4 people can be selected from a group of 10 people, we can use the concept of combinations. In combinations, the order in which the individuals are selected does not matter.
To calculate the number of ways to choose the committee, we can use the formula for combinations:
nCr = n! / (r!(n-r)!)
Where n represents the number of people to choose from (10 in this case), and r represents the number of people to be selected for the committee (4 in this case).
Using the formula, we can calculate the number of ways as follows:
10C4 = 10! / (4!(10-4)!)
= 10! / (4!6!)
= (10 * 9 * 8 * 7 * 6!) / (4 * 3 * 2 * 1 * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
Therefore, there are 210 ways a committee of 4 people can be selected from a group of 10 people.
To find the number of ways to elect positions in a club or select members for a committee, you can use the concept of combinations and permutations.
1. For the club positions:
To elect a president, secretary, and treasurer from a group of 20 people, we need to find the number of permutations.
We have 20 choices for the first position, 19 choices for the second position (as one person can hold multiple positions), and 18 choices for the third position.
Therefore, the total number of ways to elect the club positions is calculated as:
20 * 19 * 18 = 6,840 ways.
2. For the committee selection:
To select a committee of 4 people from a group of 10 people, we need to find the number of combinations.
The number of combinations can be determined using the formula for combinations, which is nCr = n! / (r! * (n - r)!)
In our case, n is 10 (total number of people) and r is 4 (number of people to be selected).
Therefore, the total number of ways to select the committee can be calculated as:
10! / (4! * (10 - 4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210 ways.
So, there are 6,840 ways to elect the club positions and 210 ways to select the committee.