`text(Permutation : )`
Permutation means arrangement in a definte order of things which may be alike or different taken some or all at a time. Hence permutation refers to the situation where order of occurrence of the events is important. or we can say each of the different arrangements which can be made by taking some or all of a number of things is called a permutation. In permutation, order of the arrangement is important.
(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.
``text(important Result: )
1. The number of permutations of n different things, taking r at a time is denoted by `text()^nP_r` or `P (n, r)` or `A (n, r ),` then
`text()^nP_r = n(n-1)(n-2)............(n-r+1)`
`= (n !) / ((n+r) !)`
`text(proof : )`
LHS = `text()^nP_r` = Number of ways of filling up r vacant places simultaneously from n different things
`tt((1,2,3,...,r),(downarrow,downarrow,downarrow,,downarrow),(n text(ways),(n-1) text(ways),(n-2) text(ways),,(n-r+1) text(ways)))`
`= n (n- 1)(n - 2) ... (n- r + 1) `
`(n (n- 1)(n- 2) ... (n- r + 1) x (n- r) )/((n-r)!) = (n !)/ ((n-r) !)`
`= R.HS`
`text(Note :)` (i) The number of permutations of n different things taken all at a time
`text( )^nP_n = n!`
`(ii) text( )^nP_0=1,text( )^nP_1 ` and `text( )^nP(n-1) = text( )^nP_n = n!`
`(iii) text( )^nP_r = n(text( )^(n-1)P_(r-1)) = n(n-1) text( )^(n-2)P_(n-2)`
`= n(n-1)(n-2)text( )^(n-3)P_(r-3)= ........`
`(iv) text( )^(n-1)P_r = (n-r)text( )^(n-1)P_(r-1)`
`(v) (text( )^nP_r)/(text( )^nP_(r-1)) = (n-r+1)`
`text(Permutation : )`
Permutation means arrangement in a definte order of things which may be alike or different taken some or all at a time. Hence permutation refers to the situation where order of occurrence of the events is important. or we can say each of the different arrangements which can be made by taking some or all of a number of things is called a permutation. In permutation, order of the arrangement is important.
(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.
``text(important Result: )
1. The number of permutations of n different things, taking r at a time is denoted by `text()^nP_r` or `P (n, r)` or `A (n, r ),` then
`text()^nP_r = n(n-1)(n-2)............(n-r+1)`
`= (n !) / ((n+r) !)`
`text(proof : )`
LHS = `text()^nP_r` = Number of ways of filling up r vacant places simultaneously from n different things
`tt((1,2,3,...,r),(downarrow,downarrow,downarrow,,downarrow),(n text(ways),(n-1) text(ways),(n-2) text(ways),,(n-r+1) text(ways)))`
`= n (n- 1)(n - 2) ... (n- r + 1) `
`(n (n- 1)(n- 2) ... (n- r + 1) x (n- r) )/((n-r)!) = (n !)/ ((n-r) !)`
`= R.HS`
`text(Note :)` (i) The number of permutations of n different things taken all at a time
`text( )^nP_n = n!`
`(ii) text( )^nP_0=1,text( )^nP_1 ` and `text( )^nP(n-1) = text( )^nP_n = n!`
`(iii) text( )^nP_r = n(text( )^(n-1)P_(r-1)) = n(n-1) text( )^(n-2)P_(n-2)`
`= n(n-1)(n-2)text( )^(n-3)P_(r-3)= ........`
`(iv) text( )^(n-1)P_r = (n-r)text( )^(n-1)P_(r-1)`
`(v) (text( )^nP_r)/(text( )^nP_(r-1)) = (n-r+1)`