Understanding Combinations
In combinatorics, a combination is a selection of items from a larger set such that the order of selection does not matter. This is distinct from a permutation, where the order is significant.
The formula for choosing $r$ items from a set of $n$ items is given by:
$$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$
The Problem Statement
We need to form a committee of 4 people from a group of 10, with the constraint that one specific person must be included.
Step-by-Step Solution
- Identify the constraints: We have a total of 10 people and need a committee of 4.
- Handle the requirement: Since one specific person must be on the committee, we have already filled 1 of the 4 available spots.
- Adjust the remaining variables:
- We now need to fill $4 - 1 = 3$ spots.
- We are choosing from $10 - 1 = 9$ remaining people.
- Calculate: We need to calculate the number of ways to choose 3 people from the remaining 9, which is $C(9, 3)$.
$$\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1}$$
$$\binom{9}{3} = 3 \times 4 \times 7 = 84$$
Conclusion
There are 84 ways to form the committee given the restriction.