The number of combinations of `n` different things taken `r` at a time is denoted by `text()^nC_r` or `C(n, r)` or `(n/r)`
Then,
`text()^nC_r=(n!)/(r!.(n-r)!)`


Note the following facts : `(i)` `text()^nC_r ` is a natural number .

`(ii)` `text()^nC_r =0`, if `r > n`

`(iii)` `text()^nC_0=text()^nC_n=1` and `text()^nC_1=n`

`(iv)` `text()^nP_r=text()^nC_r ,` if `r=0` or `1`

`(v)` `text()^nC_r=text()^nC_(n-r) ,` if ` r> n/2`

`(vi)` `text()^nC_r+text()^nC_(r-1)=text()^(n+1)C_r ` `[ text(Pascal's Rule)]`

`(vii)` if `text()^nC_x=text()^nC_y =>x=y` or `x+y=n`

`(viii)` `text()^nC_r=n/r. text()^(n-1)C_(r-1)`

`(ix)` `n.text()^(n-1)C_(r-1)=(n-r+1). text()^nC_(r-1)`

`(x)` `(text()^nC_r)/(text()^nC_(r-1))=(n-r+1)/r`

`(x i)` `(a)` if `n` is even , `text()^nC_r` is greatest for `r=n/2`

`quad quad quad quad (b)` if `n` is odd , `text()^nC_r` is greatest for `r=(n-1)/2` or `(n+1)/2`

`(x ii)` `text()^nC_0+text()^nC_1+text()^nC_2+.....................+text()^nC_n=2^n`

`(x iii)` `text()^nC_0+text()^nC_2+text()^nC_4+................=text()^nC_1+text()^nC_3+text()^nC_5+.................=2^(n-1)`

`(x iv)` `text()^(2n+1)C_0+text()^(2n+1)C_1+text()^(2n+1)C_2+...........+text()^(2n+1)C_n= 2^(2n)`

`(x iv)` `text()^nC_n+text()^(n+1)C_n+text()^(n+3)C_n +.................+ text()^(2n-1)C_n = text()^(2n)C_(n+1)`