`text(Combination : )`
(A) Combination / selection / collection/ committee refers to the situation where order of occurrence of the event is not important. Combination is selection of one or more things out of n things which may be alike or different taken some or all at a time.
Example :
(i) Out of `A, B, C, D` take `3` letters & form number plate of car. [Pemutation]
(ii) Out of four letters `A, B, C, D` take any `3` letters & form triangle (possible). [Combination]
In `1^(st)` case arrangement of letters are there, in `2^(nd)` case only selection will form the triangle, arrangment is not required.
(B) Number of combination / selection of `n` distinct things taken `r` at a time.
`text()^nC_r=C(n,r)=(n/r)=(n!)/((r)!(n-r)!)`
Proof :
Let `10` different objects are given as `A, B, C, D, E, F, G, H, I, J`
Let combinations taking `3` at a time `= x`
Arrangement `= (x) xx (3!)`
`x . 3! -text()^(10)P_3`
`x=(text()^10 P_3)/(3!)=(10!)/((10-3)3!)`
`text(Note the following facts: )`
(C)Number of combination of `n` different things taken rat a time when `p` particular things are always included.
`= text()^(n- p)C_( r-p)`
i.e. Find total number of ways of selecting `11` player out of `15` player when Mahendra Sing Dhoni and Yuvraj Singh are always included
`= text()^(15-2)C_(11-2)=text()^13C_9`
Theorm-4 :
Number of combination of `n` different things taken rat a time when `p` particular thing are always excluded.
`=text()^(n- p)C_r`
Eg: How many different selections of `6` books can be made from `11` different books if two particular books are never selected
`=text()^(11-12) C_6=text()^9C_6`
`text(Combination : )`
(A) Combination / selection / collection/ committee refers to the situation where order of occurrence of the event is not important. Combination is selection of one or more things out of n things which may be alike or different taken some or all at a time.
Example :
(i) Out of `A, B, C, D` take `3` letters & form number plate of car. [Pemutation]
(ii) Out of four letters `A, B, C, D` take any `3` letters & form triangle (possible). [Combination]
In `1^(st)` case arrangement of letters are there, in `2^(nd)` case only selection will form the triangle, arrangment is not required.
(B) Number of combination / selection of `n` distinct things taken `r` at a time.
`text()^nC_r=C(n,r)=(n/r)=(n!)/((r)!(n-r)!)`
Proof :
Let `10` different objects are given as `A, B, C, D, E, F, G, H, I, J`
Let combinations taking `3` at a time `= x`
Arrangement `= (x) xx (3!)`
`x . 3! -text()^(10)P_3`
`x=(text()^10 P_3)/(3!)=(10!)/((10-3)3!)`
`text(Note the following facts: )`
(C)Number of combination of `n` different things taken rat a time when `p` particular things are always included.
`= text()^(n- p)C_( r-p)`
i.e. Find total number of ways of selecting `11` player out of `15` player when Mahendra Sing Dhoni and Yuvraj Singh are always included
`= text()^(15-2)C_(11-2)=text()^13C_9`
Theorm-4 :
Number of combination of `n` different things taken rat a time when `p` particular thing are always excluded.
`=text()^(n- p)C_r`
Eg: How many different selections of `6` books can be made from `11` different books if two particular books are never selected
`=text()^(11-12) C_6=text()^9C_6`