Statistics Combinations

By Author: Pierce Brosnan
Total Articles: 127
Comment this article

Introduction of statistics combinations:

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 Statistics Combinations:

Think the number of provision of eight different books on a shelf. We have seen that the number of permutations of the seven books is

`7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1`=5040

However, if the order does not issue, there is only one combination of books. If there are eight dissimilar books available and only seven of them can be placed on the bookshelf, the total number of way in which this can be done is

`8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1`=40320

But, for each exacting set of seven books,

`7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1` permutations = one combination

i.e. for any one of the 5040 permutation there ...
... are 40320 other permutations of the same statistics combinations of eight books.

Therefore, the number of strange sets of seven books taken from the eight available books is given by

`=` Total number of permutations
Number of permutations of each set

Take a broad view this argument, we see that if we have N object from which we choose groups of P objects, the total number of likely collection (statistics combinations) is given by

`=` Number of permutations of P objects from N objects
Number of permutations of P objects among themselves
Example for Statistics Combinations:

Example 1:

To calculate total number of statistics combinations or how many different hands of eight 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 exacting set of eight cards that issue.

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

`52 xx 51 xx 50 xx 49 xx 48 xx 47 xx 46 xx 45`

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

`(52 xx 51 xx 50 xx 49 xx 48 xx 47 xx 46 xx 45)/(8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1)` =`(3.03423382 xx 10^13)/(40320)`= 752538150

Total number of statistics combinations is 752538150.

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

Example 2:

To calculate total number of statistics combinations or how many ways can six boys be chosen this leaves six more boys to be chosen from the remaining 30 boys.

Solution:

The number of ways of arranging five boys from nineteen is `30 xx 29 xx 28 xx 27 xx 26 xx 25`.

The number of ways of arranging the five among themselves is `6 xx 5 xx 4 xx 3 xx 2 xx 1`.

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

`(30 xx 29 xx 28 xx 27 xx 26 xx 25)/(6 xx 5 xx 4 xx 3 xx 2 xx 1)`= `(427518000)/(720)`= 593775

Total number of statistics combinations is 593775.

Learn more on about Sampling Bias and its Examples. Between, if you have problem on these topics Population Parameter, Please share your comments.