 |
| Combinations |
|
| The selection of a number of things taking some or all of them at a time are called combinations. |
| |
| The number of ways of selecting r things out of n dissimilar things is
denoted by C(n, r) or nCr. |
| |
| Example: |
| |
| From a class of 32 students, 4 are to be chosen for a competition. In how many ways can this be done? |
| |
| Suggested answer: |
| |
| We are to select 4 students from 32. This selection can done in |
| |
 |
| |
|
| |
The number of combinations of n dissimilar things, taken r at a time is  |
| |
|
| |
| C(n,r) is the required combination by definition. Each of these combinations consists of a group of r dissimilar things, which can be arranged among themselves in P(r,r) = r! ways. But the number of permutations of n different things taken r at a time is P(n,r). |
| |
 |
| |
| Corollary 1: |
| |
| C(n,r) = C(n,n-r) |
| |
| Proof: |
| |
 |
| |
| Corollary 2: |
| |
| C(n,0) = C(n,n) = 1 |
| |
| Proof: |
| |
 |
| |
| Corollary 3: |
| |
 |
| |
| Proof: |
| |
| If r = s, there is nothing to prove. |
| |
 |
| |
| Now, |
| |
 |
| |
 |
| |
| If r < s, then n - r > n - s, then the above equation becomes |
| |
 |
| |
| Since both sides are products of (s-r), consecutive integers in |
| |
 |
| |
| Similarly it can be proved that n = r + s if r > s. |
| |
 |
| |
| Corollary 4: |
| |
 |
| |
| Proof: |
| |
 |
| |
 |
| |
 |
| |
 |
| |
 |
| |
| Prove the following statements |
| |
 |
| |
 |
| |
 |
| |
 |
| |
| Proof: |
| |
 |
| |
 |
| |
 |
| |
 |
| |
 |
| |
 |
| |
 |
| |
| Difference between a Permutation and a Combination |
| |
| i. In a combination, only selection is made. In a permutation, not only a selection is made, but also there is an arrangement of a definite order. |
| |
| ii. There is no order of selection in combinations. In permutation, order is a must. |
| |
| iii. Usually (i.e., except in special cases or trivial cases), the number of permutations exceeds the number of combinations. |
| |
