MATHCOUNTS TOOLBOX

Facts, Formulas and Tricks

Lesson 10: Combinations

When different orderings are not to be counted separately, i.e. the

outcome, mn is equivalent to the outcome nm, the problem involves

combinations.

Combination Formula:Different orders of the same items are not

counted.  The combination formula is equivalent to dividing the corresponding

number of permutations by r!.n: number of available items or choices

r: the number of items to be selected    Sometimes this formula is written:

C(n,r).

Combination Formula:Different orders of the same items are not

counted.  The combination formula is equivalent to dividing the corresponding

number of permutations by r!.n: number of available items or choices

r: the number of items to be selected    Sometimes this formula is written:

C(n,r).

Taking the letters a, b, and c taken two at a time, there are six permutations: {ab, ac, ba,

bc, ca, cb}.  If the order of the arrangement is not important, how many of these outcomes are equivalent, i.e. how many combinations

are there?

Taking the letters a, b, and c taken two at a time, there are six permutations: {ab, ac, ba,

bc, ca, cb}.  If the order of the arrangement is not important, how many of these outcomes are equivalent, i.e. how many combinations

are there? ab = ba; ac = ca; and bc = cb

The three duplicate permutations would not be counted, therefore three

combinations exist

Calculate the value of 7C4.

Calculate the value of 7C4.

This represents a combination of 7 objects

taken 4 at a time and is equal to

Calculate the value of 7C4.

This represents a combination of 7 objects

taken 4 at a time and is equal to

Calculate the value of 9C5

Calculate the value of 9C5

This represents a combination of 9 objects taken 5 at a time and is

equal to . . .

Calculate the value of 9C5

This represents a combination of 9 objects taken 5 at a time and is

equal to . . .

In how many ways can three class representatives be chosen from a group of twelve students?  If the order of the

arrangement is not important, how many outcomes will there be? 

In how many ways can three class representatives be chosen from a group of twelve students?  If the order of the

arrangement is not important, how many outcomes will there be? 

This represents a combination of 12 objects taken 3 at a time and is equal to

In how many ways can three class representatives be chosen from a group of twelve students?  If the order of the

arrangement is not important, how many outcomes will there be? 

This represents a combination of 12 objects taken 3 at a time and is equal to

Fini!