To determine how many different committees of 7 people can be formed from a group of 10 people, we can use the concept of combinations from combinatorial mathematics. The formula for combinations is given by:
C(n, r) = n! / (r! * (n – r)!)
Where:
- n is the total number of items to choose from (in this case, 10 people).
- r is the number of items to choose (in this case, 7 people).
- ! denotes factorial, which is the product of all positive integers up to that number.
Applying the values for our specific scenario:
- n = 10
- r = 7
So, we need to calculate:
C(10, 7) = 10! / (7! * (10 – 7)!)
This simplifies to:
C(10, 7) = 10! / (7! * 3!)
Next, we can break this down further:
10! = 10 × 9 × 8 × 7! (we can cancel the 7!)
So, we have:
C(10, 7) = (10 × 9 × 8) / 3!
Calculating 3!:
3! = 3 × 2 × 1 = 6
Now, substituting back into our equation:
C(10, 7) = (10 × 9 × 8) / 6
Calculating the numerator:
10 × 9 × 8 = 720
Now, dividing by 6:
C(10, 7) = 720 / 6 = 120
Therefore, the total number of different committees of 7 people that can be formed from a group of 10 people is 120.