Permutation and combination are both concepts in combinatorics, a branch of mathematics concerned with counting and arranging objects.
**Permutation:**
A permutation is an arrangement of objects in a specific order. It refers to the different ways in which a set of objects can be ordered or arranged. The number of permutations of \( n \) objects taken \( r \) at a time is denoted as \( P(n, r) \) and calculated using the formula:
\[ P(n, r) = \frac{{n!}}{{(n - r)!}} \]
where:
- \( n! \) (read as "n factorial") represents the product of all positive integers from 1 to \( n \),
- \( (n - r)! \) represents the product of all positive integers from 1 to \( n - r \).
For example, if you have 5 different books and you want to arrange 3 of them on a shelf in a specific order, the number of permutations would be \( P(5, 3) = \frac{{5!}}{{(5 - 3)!}} = \frac{{5!}}{{2!}} = 5 \times 4 \times 3 = 60 \).
**Combination:**
A combination is a selection of objects from a set without regard to the order in which they are arranged. It refers to the different ways in which a subset of objects can be chosen from a larger set. The number of combinations of \( n \) objects taken \( r \) at a time is denoted as \( C(n, r) \) or \( \binom{n}{r} \) and calculated using the formula:
\[ C(n, r) = \binom{n}{r} = \frac{{n!}}{{r! \times (n - r)!}} \]
where:
- \( n! \) is the factorial of \( n \),
- \( r! \) is the factorial of \( r \),
- \( (n - r)! \) is the factorial of \( n - r \).
For example, if you have 5 different books and you want to choose 3 of them to take on a trip without considering the order, the number of combinations would be \( C(5, 3) = \binom{5}{3} = \frac{{5!}}{{3! \times (5 - 3)!}} = \frac{{5!}}{{3! \times 2!}} = 10 \).
In summary, permutations deal with arrangements or orderings, while combinations deal with selections or subsets without regard to order. Both concepts are widely used in various areas of mathematics, including probability, statistics, and combinatorial analysis.