4. Combinations (Unordered Selections)

A combination of n objects taken r at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important.

Example 1

Consider the selection of a set of 4 different letters from the English alphabet.

Suppose

Note: David and John selected the same set of letters, even though they selected them in different order. Hence, these 3 people have selected only 2 different sets of 4 letters (not 3 sets!!).

Question: How many different sets of 4 letters can be selected from the alphabet?

Using the result from the above example and generalising, we have the following expression for combinations.

Number of Combinations

The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by:

`C_r^n=(n!)/(r!(n-r)!`

Notes

(a) `C_r^n=(P_r^n)/(r!)`

(b) `C_0^n=1`

(c) `C_n^n=1`

(d) `C_r^n=C_(n-r)^n`

Expand each one out from their definitions to understand why they work.

In our example above, the number of different sets of `4` letters which can be chosen from the alphabet is

`C_4^26` `=(26!)/(4!(26-4)!)` `=(26!)/(4!22!)` `=14950`

Example 2

Find the number of ways in which `3` components can be selected from a batch of `20` different components.

Example 3

In how many ways can a group of `4` boys be selected from `10` if

(a) the eldest boy is included in each group?

(b) the eldest boy is excluded?

(c) What proportion of all possible groups contain the eldest boy?

Example 4

A class consists of `15` boys of whom `5` are prefects.

How many committees of `8` can be formed if each consists of

(a) exactly `2` prefects? (b) at least `2` prefects?

Example 5

Out of `5` mathematicians and `7` engineers, a committee consisting of `2` mathematicians and `3` engineers is to be formed. In how many ways can this be done if

(a) any mathematician and any engineer can be included?

(b) one particular engineer must be in the committee?

(c) two particular mathematicians cannot be in the committee?

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