`=>` Let us now assume that there is a group of `3` lawn tennis players `color(blue)(X, Y, Z.)` A team consisting of `2` players is to be formed.

`color(blue)("In how many ways can we do so ?")`

Here, `color(blue)("order is not important.")` In fact, there are only `3` possible ways in which the team could be constructed.

These are `color(blue)(XY, YZ)` and `color(blue)(ZX)` (Fig 1. )

Here, each selection is called a combination of `3` different objects taken `2` at a time. In a combination, the order is not important.

Some more illustrations where order is not important

Twelve persons meet in a room and each shakes hand with all the others. How do we determine the number of hand shakes. X shaking hands with Y and Y with X will not be two different hand shakes. Here, order is not important. There will be as many hand shakes as there are combinations of 12 different things taken 2 at a time.

Seven points lie on a circle. How many chords can be drawn by joining thesepoints pairwise? There will be as many chords as there are combinations of 7 different things taken 2 at a time.

Now, we obtain the formula for finding the number of combinations of n different objects taken r at a time, denoted by `color(red)(text()^nC_r).`

`=>` `color(green)("Suppose we have 4 different objects A, B, C and D. Taking 2 at a time")`,

if we have to make combinations, these will be `AB, AC, AD, BC, BD, CD.`

Here, `AB` and `BA` are the same combination as order does not alter the combination.

This is why we have not included `BA, CA, DA, CB, DB` and `DC` in this list.

There are as many as `6` combinations of `4` different objects taken `2` at a time, i.e., `text()^4C_2 = 6.`

Corresponding to each combination in the list, we can arrive at `2!` permutations as `2` objects in each combination can be rearranged in `2!` ways. Hence, the number of permutations `= text()^4C_2 × 2!.`

On the other hand, the number of permutations of `4` different things taken `2` at a time `= text()^4P_2.`

Therefore `text()^4P_2 = text()^4C_2 × 2!` or `(4!)/((4-2)! 2!)= text()^4C_2 `
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" `

`color(red)(★ ul"Combinations")`

Combination is selection of one or more things out of n things which may be alike or different taken some oral! at a time.

it is denoted by `color(red)(text()^nC_r).`

` color(blue)"Theorem - "` `color(red)(text()^nP_r = text()^nC_r × r! , \ \ \ \ \ 0 < r ≤ n.)`

`"Proof"` Corresponding to each combination of `text()^nC_r ` we have `r !` permutations,

because `r` objects in every combination can be rearranged in `r !` ways.

Hence, the total number of permutations of `n` different things taken `r` at a time is `text()^nC_r × r! ,` On the other hand, it is `text()^nP_r .`

Thus `text()^nP_r = text()^nC_r × r! , \ \ \ \ \ 0 < r ≤ n.`


`\color{green} ★ \color{green} \mathbf(KEY \ POINTS)`

`\color{green} (1-)` From above ` (n!)/( ( n-r)!) = text()^(n)C_r xx r! ` , i.e., `text()^(n)C_r = (n!)/(r! ( n-r)! )`

In particular, if `r = n` , `color(red)(text()^nC_n = (n!)/(n! 0!) =1)` .

`\color{green} (2-)` As `(n!)/(0! (n-0)! ) =color(red)(1 = text()^(n)C_0)`, the formula `text()^nC_r = (n!)/( (n-r)! r!) ` is applicable for r = 0 also.

Hence

`color(red)(text()^(n)C_r = (n!)/( (n-r)! r!)) , 0 le r le n` .

`\color{green} (3-)` `text()^(n)C_(n-r) = (n!)/( (n-r)! ( n - ( n-r) )! ) = (n!)/( (n-r)! r!) = text()^(n)C_r` .,

i.e., selecting r objects out of n objects is same as rejecting` (n – r)` objects.

`\color{green} (4-)` `color(red)(text()^nC_a = text()^(n)C_b) => a =b ` or `a = n - b` , i.e., `color(blue)(n =a + b)`

`\color { maroon} ® \color{maroon} ul (" REMEMBER") `

`color(red)(text()^nC_r +text()^nC_(r-1)=text()^(n+1)C_r ) `

`"Proof"` We have : `text()^nC_r+text()^nC_(r-1) = (n!)/(r! (n- r)!) + ( n!)/((r-1)! ( n-r+1)!)`

` = (n!)/(r xx (r-1)! ( n-r)!) + (n!)/((r-1) ! ( n-r+1)(n-r)!)`

`= (n!)/((r-1)! (n-r)!) [1/r +1/(n-r+1)]`

` = (n!)/((r-1) !(n-r)!) xx (n -r +1+r)/(r(n-r+1)) = ((n+1)!)/(r!(n+1-r)!) = text()^(n+1)C_r`

Combination/selection/collection/committee/team refers to the situation where order of occurrence of the
event is not important.