To determine how many ways we can choose a committee of 3 people from a group of 5, we can use the concept of combinations in combinatorics. Combinations are used when the order of selection does not matter.
The formula for combinations is given by:
C(n, r) = n! / (r!(n – r)!)
Where:
- C(n, r) is the number of combinations of n items taken r at a time.
- n is the total number of items.
- r is the number of items to choose.
- ! denotes factorial, which is the product of all positive integers up to that number.
In this scenario:
- n = 5 (the total number of people),
- r = 3 (the number of people to choose).
Substituting these values into the formula:
C(5, 3) = 5! / (3!(5 – 3)!)
This simplifies to:
C(5, 3) = 5! / (3! imes 2!)
Calculating the factorials:
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 3! = 3 × 2 × 1 = 6
- 2! = 2 × 1 = 2
Now plug these values back into the equation:
C(5, 3) = 120 / (6 × 2) = 120 / 12 = 10
Therefore, there are a total of 10 different ways to form a committee of 3 people from a group of 5 individuals.