A school has three grade 10 classes. In how many ways can 3 out of 8 mathematics teachers be allocated to the classes?
What is 8 C 3
or
C(8,3) ?
- most up-to-date calculators have an n C r
as a function.
To find the number of ways to allocate 3 mathematics teachers to the three grade 10 classes, we can use the concept of combinations.
Combinations are used when the order of selection does not matter. In this case, since we are only concerned with which teachers go to which classes and not the specific order in which they are chosen, we need to use combinations.
The formula to calculate combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
Where:
- n is the total number of items (in this case, the number of mathematics teachers, which is 8)
- r is the number of items to choose (in this case, the number of mathematics teachers we want to allocate to the classes, which is 3)
- ! denotes the factorial, which is the product of all positive integers less than or equal to a given number.
Using the combination formula, we can calculate the number of ways to allocate 3 out of 8 mathematics teachers to the classes as follows:
C(8, 3) = 8! / (3! * (8 - 3)!)
= 8! / (3! * 5!)
Now, let's calculate the factorial values:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
3! = 3 * 2 * 1 = 6
5! = 5 * 4 * 3 * 2 * 1 = 120
So, plugging these values into the formula:
C(8, 3) = 40,320 / (6 * 120)
= 40,320 / 720
= 56
Therefore, there are 56 ways to allocate 3 mathematics teachers to the three grade 10 classes.