How many different groups of 7 students can you create from a class of 20 students?
How many different results could there be for 10 political candidates running for the same office in the upcoming election?
number different? 20!/7!13!
There can only be 10 different results, only one person can win
There can only be 10 different results, only one person can win.
how would i set this up?
To determine the number of different groups of 7 students that can be created from a class of 20 students, you can use the concept of combinations. The formula for combinations is given by:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of items and r is the number of items chosen at a time.
For the first question, you have n = 20 (the total number of students) and r = 7 (the number of students to be chosen). Plugging these values into the formula:
C(20, 7) = 20! / (7! * (20-7)!)
Simplifying further, this becomes:
C(20, 7) = 20! / (7! * 13!)
Calculating the factorial values:
20! = 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13!
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
We can cancel out the common terms:
C(20, 7) = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13!) / (7 * 6 * 5 * 4 * 3 * 2 * 1 * 13!)
Cancelling out the common terms, we get:
C(20, 7) = (20 * 19 * 18 * 17 * 16 * 15 * 14) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
Calculating the expression, the number of different groups of 7 students that can be created from a class of 20 students is:
C(20, 7) = 77520
Now, let's move on to the second question.
To determine the number of different results for 10 political candidates running for the same office in the upcoming election, you can use the concept of permutations. The formula for permutations is given by:
P(n, r) = n! / (n-r)!
where n is the total number of items and r is the number of items chosen at a time.
In this case, you have n = 10 (the total number of candidates) and r = 10 (all candidates running for the office). Plugging these values into the formula:
P(10, 10) = 10! / (10-10)!
Simplifying further, this becomes:
P(10, 10) = 10! / 0!
Now, since the factorial of 0 is defined as 1:
P(10, 10) = 10! / 1
Calculating the factorial value:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
The number of different results for 10 political candidates running for the same office is:
P(10, 10) = 10! = 3,628,800