Indistinguishable objects

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Dokumen menjelaskan tentang objek yang tidak semuanya berbeda, dengan contoh kelereng yang terdiri dari 2 jenis warna hitam dan putih. Diberikan rumus untuk menghitung permutasi benda yang tidak semuanya berbeda, yaitu n!/(r!(n-r)!) untuk r jenis pertama dan n-r jenis lainnya, serta rumus permutasi benda dengan lebih dari 2 jenis yaitu n!/(r1!r2!...rk!) unt

INDISTINGUISHABLE OBJECTS
Adalah objek atau benda yang tidak
semuanya berbeda

Muncul saat kita akan mengetahui,
bagaimana banyaknya kemungkinan
permutasi bila beberapa benda dalam
suatu himpunan sama?

Contoh 1.6.11
Terdapat 5 kelereng, 2 hitam,3 putih,
tetapi tidak semua berbeda. Akan
ditunjukan susunan objek yang berbeda
dari 2 kelereng hitam dan dua kelereng
putih.

Susunan 5 objek yang berbeda dari 2 tipe
pertama dan 3 tipe yang lain.
BBWWW WWWBB
BWBWW BWWBW
WBWWB WWBWB
WBWBW BWWWB
WWBBW WBBWW

Kita mengartikan 10 susunan yang berbeda
tadi sebagai permutasi 5 objek yang tidak
semuanya berbeda sehingga jumlah permutasi
benda yang tidak semuanya berbeda adalah

5!
2!3!

Theorema 1.6.5

Banyaknya permutasi dari n objek. Bila
sekumpulan n objek beranggotakan r jenis
pertama dan n-r jenis lainnya adalah :

n!  n  n 
= =
 r   n − r

r !( n − r ) !    

Theorema 1.6.6

Banyaknya permutasi n benda, bila
sekumpulan n objek beranggotakan r1
jenis pertama, r2 jenis kedua,….. rk jenis ke
k adalah :
n!
r1 !r2 !.....rk !
r1 +r2+……+ rk = n

Contoh 1.6.12

Terdapat 10 kelereng, 2 hitam, 3 putih, dan
5 merah, tetapi tidak semua berbeda.
Maka banyaknya permutasi adalah :

10!
= 2520
2!3!5!

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Indistinguishable objects

1. INDISTINGUISHABLE OBJECTS Adalah objek atau benda yang tidak semuanya berbeda Muncul saat kita akan mengetahui, bagaimana banyaknya kemungkinan permutasi bila beberapa benda dalam suatu himpunan sama?

2. Contoh 1.6.11 Terdapat 5 kelereng, 2 hitam,3 putih, tetapi tidak semua berbeda. Akan ditunjukan susunan objek yang berbeda dari 2 kelereng hitam dan dua kelereng putih.

3. Susunan 5 objek yang berbeda dari 2 tipe pertama dan 3 tipe yang lain. BBWWW WWWBB BWBWW BWWBW WBWWB WWBWB WBWBW BWWWB WWBBW WBBWW

4. Kita mengartikan 10 susunan yang berbeda tadi sebagai permutasi 5 objek yang tidak semuanya berbeda sehingga jumlah permutasi benda yang tidak semuanya berbeda adalah 5! 2!3!

5. Theorema 1.6.5 Banyaknya permutasi dari n objek. Bila sekumpulan n objek beranggotakan r jenis pertama dan n-r jenis lainnya adalah : n!  n  n  = =  r   n − r  r !( n − r ) !    

6. Theorema 1.6.6 Banyaknya permutasi n benda, bila sekumpulan n objek beranggotakan r1 jenis pertama, r2 jenis kedua,….. rk jenis ke k adalah : n! r1 !r2 !.....rk ! r1 +r2+……+ rk = n

7. Contoh 1.6.12 Terdapat 10 kelereng, 2 hitam, 3 putih, dan 5 merah, tetapi tidak semua berbeda. Maka banyaknya permutasi adalah : 10! = 2520 2!3!5!

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Indistinguishable objects

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Indistinguishable objects