Difference between Combinations and Permutations

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. For example AB and BA are same as combinations, but different as permutations.

Associate the word selection for combinations and arrangement for permuations.

**Notation**

The number of all combinations of n objects, taken r at a time is denoted by C(n,r) or

^{n}C

_{r}.

^{n}C

_{r}is defined when n and r are non-negative numbers.

**Theorem**

The number of all combinations of n distinct objects, taken r at a time is given by

^{n}C

_{r}= n!/(n-r)!r!

Results from the theorem

^{n}C

_{r}= [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)

^{n}C

_{n}=1

^{n}C

_{0}= 1

^{n}C

_{r}=

^{n}P

_{r}/r!

Properties of

^{n}C

_{r}and C(n,r)

1.

^{n}C

_{r}=

^{n}C

_{n-r}

Note: If x=y = n

^{n}C

_{x}=

^{n}C

_{y}

2. Let n and r be non-negative integers such that r≤n. Then

^{n}C

_{r}+

^{n}C

_{r-1}=

^{n+1}C

_{r}

3. Let n and r be non-negative integers such that 1≤ r≤n. Then

^{n}C

_{r}= (n/r)

^{n-1}C

_{r-1}

4. If 1≤ r≤n, then

n.

^{n-1}C

_{r-1}= (n-r+1)

^{n}C

_{r-1}

5.

^{n}C

_{x}=

^{n}C

_{y}implies x+y = n

6. If n is even, then the greatest value of

^{n}C

_{r}[0≤ r≤n] is

^{n}C

_{n/2}.

7. If n is odd, then the greatest value of

^{n}C

_{r}[0≤ r≤n] is

^{n}C

_{(n+1)/2}or

^{n}C

_{(n-1)/2}.

**Selection of one or more items**

Selection from different items

The number of ways of selecting one or more items from a group of n distinct items is 2ⁿ - 1.

Selection from identical items

1. The number of ways of selecting r items out of n identical items is 1.

2. The total number of ways of selecting zero or more i.e. at least one item from a group of n identical items is (n+1).

3. The total number of selections of some or all out of p+q+r items where p are alike of one kind, q are alike of second kind, and rest are alike of third kind is {(p+1)(q+1)(r+1)}-1.

Selection of items from a group containing both identical and different items

1. the total number of ways of selecting one or more items from p identical items of one kind; q identical items of second kind, r identical items of third kind and n different items is

[(p+1)(q+1)(r+1) 2ⁿ]-1

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