A __________ is an arrangement of r objects chosen from n distinct objects without repetition and without regard to order.

Verified step by step guidance

1. Identify the key terms in the problem: 'arrangement of r objects chosen from n distinct objects without repetition and without regard to order.'
2. Understand that 'without repetition' means each object can be chosen only once, and 'without regard to order' means the order in which objects are chosen does not matter.
3. Recall the definitions of permutations and combinations: permutations consider order, combinations do not.
4. Since order does not matter here, the correct term is a 'combination.'
5. Therefore, the blank should be filled with 'combination,' which is mathematically represented as \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

Key Concepts

Combination

A combination is a selection of r objects from a set of n distinct objects where order does not matter and no object is repeated. It counts the number of ways to choose items without regard to arrangement.

Permutation

A permutation is an arrangement of r objects chosen from n distinct objects where order matters and no repetition is allowed. It differs from combinations because the sequence of objects is important.

Without Repetition

Without repetition means each object can be selected only once in the arrangement or selection. This restriction affects the total number of possible combinations or permutations.