$A, B$ αှα့် $C$ αူαုံးαောα်αှိαα် αိုαါα
ို့။ $A, B$ αှα့် $C$ αို αျα်းαြောα့် αေαာαျαားαိုα်αα့် α‘α
ီα‘α
α်αှာ $3! = 6$ ways αြα
်αြောα်း ααα် post (part 1, part 2, part 3) αို့αွα် αα်αြαဲ့αြီးαြα
်αα်။
$A, B$ αှα့် $C$ αို αျα်းαြောα့် αေαာαျαားαိုα်αα့် α‘α
ီα‘α
α် (α) αုαှာ ABC, BCA, CAB, ACB, BAC, CBA αို့ αြα
်αα်။
αို့αာαွα် α‘αိုαါαူαုံးα¦းαို α
ားαွဲαိုα်းαွα် αေαာαျαားαα့်α‘αါ ABC, BCA, CAB α‘α
ီα‘α
α်αုံးαုαှာ α‘αူαူαα်αြα
်αြောα်း αွေ့ααα်။
αို့αြောα့် αုံးαောα်αွα် αααα¦းαုံးαူ၏ αေαာαျαားαှုα‘α
ီα‘α
α်αို αα့်αွα်းαွα်αျα်αα် ααိုαောαြောα့် $(3 - 1)! = 2! = 2$ ways αာ αြα
်αα်။
αို့αြောα့် α
α်αိုα်းαုံ αα်းαြောα်းαေါ်αွα် ααူαီαော α‘αာαα္αု $n$ αို α
ီα
α်αိုα်αော αα်းαα်းα‘αေα‘αွα်αှာ $(n - 1)!$ αြα
်αα်။
α€αွα် αα်αာαα
် (anticlockwise direction) αှα့် αα်αဲαα
် (clockwise direction) αို့αို ααူαီαော α‘α
ီα‘α
α်αျား α‘αြα
်αα်αှα်αါαα်။ α‘αα်၍ αα်αာαα
် αှα့် αα်αဲαα
် αို့αα် αူαီαော α‘αြေα‘αေ (α₯ααာ - αုαီး) αွα် α‘αာαα္αု $n$ αို α
ီα
α်αိုα်αော αα်းαα်း α‘αေα‘αွα်αှာ $\displaystyle\frac{(n - 1)!}{2}$ αြα
်αα်။
CIRCULAR PERMUTATION
If $n$ different things can be arranged in a row, the linear arrangements is $n!$, whereas every linear arrangements have a beginning and end but in circular permutations, there is neither beginning nor end.
When clockwise and anti-clockwise orders are taken as different, the number of circular permutations of $n$ different things taken all at a time is
$\begin{array}{|c|}\hline(n – 1)!\\ \hline\end{array}$
But, when the clockwise and anti-clockwise orders are not different, i.e. the arrangements of beads in a necklace, arrangements of flowers in a garland, etc.
The number of circular permutations of $n$ different things is
RESTRICTED CIRCULAR PERMUTATIONS
If clockwise and anti-clockwise arrangements are taken as different, the number of circular permutations of $n$ different things, taken $r$ at a time is given by
$\begin{array}{|c|}\hline\displaystyle\frac{{}^nP_{r}}{r}\\ \hline\end{array}$
If clockwise and anti-clockwise arrangements are taken as different, the number of circular permutations of $n$ different things, taken $r$ at a time is given by
$\begin{array}{|c|}\hline\displaystyle\frac{{}^nP_{r}}{2r}\\ \hline\end{array}$
α‘ောα်αါ video αှα့် αှα်αွဲαေ့αာαြα့်αါ။
Video Credit : Don't Memorise YouTube Channel
| Example (1) At a dinner party 3 men and 3 women sit at a round table. In how many ways can they sit if: (a) there are no restrictions? (b) men and women in alternate arrangement? (c) U Kyaw and U Myo must sit together?  (b) Number of ways = (3 – 1)! 3!= 2! 3! (c) Arrangement : (U Kyaw and U Myo) and other 4 members Number of ways = 2! 4! |
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| EXERCISES 1. In how many ways, can we arrange 6 different flowers in a circle? 2. It is decided to label the vertices of a rectangle with the letters A, B, C and D. In how many ways is this possible if: (a) they are to be in clockwise alphabetical order? (b) they are to be in alphabetical order? (c) they are to be in random order? 3. In how many ways, 6 Myanmars and 5 Koreans can be seated in a round table if (i) there is no restriction? (ii) all the 5 Koreans sit together? (iii) all the 5 Koreans do not sit together? (iv) no two Koreans sit together? 4. In how many ways, 20 persons be seated around a round table if there are 10 seats available there? |
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