Introduction to Combinations
In mathematics, a combination is a way of selecting items from a collection where the order of selection does not matter. When we need to choose $r$ items from a set of $n$ distinct items, we use the combination formula:
$$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$
Where $n!$ (n factorial) is the product of all positive integers up to $n$.
The Problem
Question: From 10 persons, in how many ways can a selection of 4 be made if two particular persons are always excluded?
Step-by-Step Solution
Analyze the constraints: We have a total of 10 people. We must choose 4, but 2 specific people are permanently excluded.
Adjust the pool size: Since 2 people are excluded, they cannot be part of our selection. We effectively remove them from the total pool. New total available people = $10 - 2 = 8$.
Identify the selection target: We still need to select a group of 4 people from the remaining pool.
Apply the formula: We need to find the number of combinations of 8 people taken 4 at a time. $$C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}$$
Calculation: $$\frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70$$
Result: There are 70 ways to make the selection.
Intuition
Think of it as simply shrinking the world you are choosing from. If you are told two people are "banned" from the team, you act as if they don't exist. You are left with 8 people to fill 4 spots. The order of these people doesn't matter, hence we use combinations, not permutations.