Calculate Combinations

Introduction:

In combinatorial mathematics, a k-combination of a finite set S is a subset of k distinct elements of S. Specifying a subset does not arrange them in a particular order; by contrast, producing the k distinct elements in a specific order defines a sequence without repetition, also called k-permutation (but which is not a permutation of S in the usual sense of that term).

Source: Wikipedia

Concept of Calculate Combinations:

Think the number of arrangements of four different books on a shelf. We have seen that the number of permutations of the four books is

3x2x1=6

However, if the order does not matter, there is only one combination of books. If there are, five different books available and only four of them can be placed on the shelf, the total number of way in which this can be done is

4x3x2=24

But, for each particular set of four books,

4 x 3 x 2 x 1 permutations = one combination

i.e. for any one of the 120 permutation there are 23 other permutations of the same combination of four books.

So the number of different sets of four books taken from the five available books is given by  
`=`  Total number of permutations   
Number of permutations of each set  

Generalizing this argument we see that if we have n object from which we select groups of p objects, the total number of possible groups(Math Combinations) is given by
`=`  Number of permutations of p objects from n objects   
Number of permutations of p objects among themselves

Example for Combination Calculator

Example 1:

To calculate combination or  how many different hands of five cards can be dealt from a pack of fifty two playing cards?

Solution:

The order in which the cards are dealt is irrelevant; it is the particular set of four cards that matters.

The number of ways of arranging four cards for fifty-two is

52*51*50*49*48

The number of ways of arranging any one set of five cards among themselves is

`(52*51*50*49*48)/(5*4*3*2*1)` =`(311875200)/(120)`= 2598960

In this following example, we investigate problems when the selection is restricted.

Example 2:

To calculate combination or  how many ways can five boys be chosen this leaves four more boys to be chosen from the remaining nineteen boys.

Solution:

The number of ways of arranging five boys from nineteen is 19*18*17*16*15.

The number of ways of arranging the five among themselves is 5*4*3*2*1.

Therefore the number of ways of choosing the five boys to join the captain is

`(19*18*17*16*15)/(5*4*3*2*1)`= `(1395360)/(120)`=11628