For AI agents: use /llms.txt for the Nakafa content index.
Permutation with identical objects is an arrangement of objects where there are several objects that are identical or the same. When there are identical objects, the number of different arrangements will decrease because exchanging identical objects does not produce new arrangements .
Imagine arranging letters from the word "MAMA". Although there are 4 4 4 M M M M M M
For permutation of n n n the formula used is:
n n n r 1 , r 2 , r 3 , … , r k r_1, r_2, r_3, \ldots, r_k r 1 , r 2 , r 3 , … , r k k k k How to identify identical objects : Count how many times each object appears in the entire arrangement, not just looking at different objects.
Let's calculate how many letter arrangements can be made from the word "KALIMANTAN".
Systematic identification steps:
Write letters one by one: K-A-L-I-M-A-N-T-A-N
Letter identification: K K K 1 time 1 \text{ time} 1 time A A A 3 times 3 \text{ times} 3 times 2 , 6 , 9 2, 6, 9 2 , 6 , 9 L L L 1 time 1 \text{ time} 1 time I I I 1 time 1 \text{ time} 1 time M M M 1 time 1 \text{ time} 1 time N N N 2 times 2 \text{ times} 2 times 7 , 10 7, 10 7 , 10 T T T 1 time 1 \text{ time} 1 time
Simplify the fraction by canceling common factors:
Divide 10 10 10 2 2 2 10 2 = 5 \frac{10}{2} = 5 2 10 = 5
So: 5 × 9 × 8 × 7 × 6 × 5 × 4 5 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 5 × 9 × 8 × 7 × 6 × 5 × 4
= 5 × 9 × 8 × 7 × 6 × 5 × 4 = 302,400 = 5 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 = 302{,}400 = 5 × 9 × 8 × 7 × 6 × 5 × 4 = 302 , 400
For the word "PALAPA" with 6 6 6
Letter identification: P P P 2 times 2 \text{ times} 2 times 1 , 5 1, 5 1 , 5 A A A 3 times 3 \text{ times} 3 times 2 , 4 , 6 2, 4, 6 2 , 4 , 6 L L L 1 time 1 \text{ time} 1 time
Calculate each factorial:
6 ! = 6 × 5 × 4 × 3 × 2 × 1 = 720 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 6 ! = 6 × 5 × 4 × 3 × 2 × 1 = 720
2 ! = 2 × 1 = 2 2! = 2 \times 1 = 2 2 ! = 2 × 1 = 2
3 ! = 3 × 2 × 1 = 6 3! = 3 \times 2 \times 1 = 6 3 ! = 3 × 2 × 1 = 6
Simplify by dividing 6 6 6 2 2 2
6 2 = 3 \frac{6}{2} = 3 2 6 = 3
So: 3 × 5 × 4 = 60 arrangements 3 \times 5 \times 4 = 60 \text{ arrangements} 3 × 5 × 4 = 60 arrangements
To solve permutation problems with identical objects, follow these steps:
Count total objects : Determine the value of n n n Identify identical objects : Group objects that are identicalCount frequency : Determine how many times each object appearsApply formula : Insert into the permutation formulaCalculate factorial : Complete the calculation carefully Let's apply these steps to find arrangements of the word "BANANA":
Count total objects
Write letters one by one: B-A-N-A-N-A
Total letters: n = 6 n = 6 n = 6
Identify identical objects
Group identical letters together:
B group: B
A group: A, A, A
N group: N, N
Count frequency
Count how many times each letter appears:
B B B 1 time 1 \text{ time} 1 time A appears 3 times 3 \text{ times} 3 times
N appears 2 times 2 \text{ times} 2 times
Apply formula
Use the permutation formula with identical objects:
Calculate factorial
Simplify the fraction first:
Therefore, the word "BANANA" can be arranged in 60 60 60
Regular permutation : All objects are different, using formula n ! n! n !
Permutation with identical objects : There are identical objects, using formula:
Arranging letters A, B, C, D (all different): 4 ! = 24 ways 4! = 24 \text{ ways} 4 ! = 24 ways
Arranging letters A , A , B , C A, A, B, C A , A , B , C
Identical objects reduce the number of arrangements because exchanging identical objects does not produce differences.
How many letter arrangements can be made from the word "MATEMATIKA"?
A flower shop has 8 8 8 3 3 3 3 3 3 2 2 2
From the digits 1 , 1 , 2 , 2 , 2 , 3 1, 1, 2, 2, 2, 3 1 , 1 , 2 , 2 , 2 , 3 6 6 6
How many different letter arrangements does the word "INDONESIA" have?
The word "MATEMATIKA" has 10 10 10
Letters one by one: M-A-T-E-M-A-T-I-K-A
Letter identification: M M M 2 times 2 \text{ times} 2 times 1 , 5 1, 5 1 , 5 A A A 3 times 3 \text{ times} 3 times 2 , 6 , 10 2, 6, 10 2 , 6 , 10 T T T 2 times 2 \text{ times} 2 times 3 , 7 3, 7 3 , 7 E E E 1 time 1 \text{ time} 1 time I I I 1 time 1 \text{ time} 1 time K K K 1 time 1 \text{ time} 1 time
Simplify the fraction by canceling common factors:
Calculate with simplification:
Complete calculation: 10 × 9 × 8 × 7 × 6 × 5 × 4 = 604,800 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 = 604{,}800 10 × 9 × 8 × 7 × 6 × 5 × 4 = 604 , 800
Divide by 4 4 4
Total 8 8 8 3 3 3 3 3 3 2 2 2
Simplify the fraction by canceling common factors:
Digits 1 , 1 , 2 , 2 , 2 , 3 1, 1, 2, 2, 2, 3 1 , 1 , 2 , 2 , 2 , 3 6 6 6
Digit identification: digit 1 1 1 2 times 2 \text{ times} 2 times 2 2 2
The word "INDONESIA" has 9 9 9
Letters one by one: I-N-D-O-N-E-S-I-A
Letter identification: I I I 2 times 2 \text{ times} 2 times 1 , 8 1, 8 1 , 8 N N N
Calculate with simplification:
Divide 6 6 6 2 2 2 6 2 = 3 \frac{6}{2} = 3 2 6 = 3
So: 3 × 5 × 4 = 60 3 \times 5 \times 4 = 60 3 × 5 × 4 = 60
604,800 4 = 151,200 arrangements \frac{604{,}800}{4} = 151{,}200 \text{ arrangements} 4 604 , 800 = 151 , 200 arrangements
Calculate with simplification:
Divide 6 6 6 12 12 12 6 12 = 1 2 \frac{6}{12} = \frac{1}{2} 12 6 = 2 1
So: 8 × 7 × 1 2 × 5 × 4 8 \times 7 \times \frac{1}{2} \times 5 \times 4 8 × 7 × 2 1 × 5 × 4
= 4 × 7 × 5 × 4 = 560 ways = 4 \times 7 \times 5 \times 4 = 560 \text{ ways} = 4 × 7 × 5 × 4 = 560 ways
3 times 3 \text{ times} 3 times 1 time 1 \text{ time} 1 time Simplify the fraction by canceling common factors:
Calculate with simplification:
Divide 6 6 6 2 2 2 6 2 = 3 \frac{6}{2} = 3 2 6 = 3
So: 3 × 5 × 4 = 60 numbers 3 \times 5 \times 4 = 60 \text{ numbers} 3 × 5 × 4 = 60 numbers
2 times 2 \text{ times} 2 times 1 time 1 \text{ time} 1 time 1 time 1 \text{ time} 1 time 1 time 1 \text{ time} 1 time 1 time 1 \text{ time} 1 time 1 time 1 \text{ time} 1 time Simplify the fraction by canceling common factors:
Calculate with simplification:
Divide by 2 2 2 8 2 = 4 \frac{8}{2} = 4 2 8 = 4
So: 9 × 4 × 7 × 6 × 5 × 4 × 3 9 \times 4 \times 7 \times 6 \times 5 \times 4 \times 3 9 × 4 × 7 × 6 × 5 × 4 × 3
= 9 × 4 × 7 × 6 × 5 × 4 × 3 = 90,720 arrangements = 9 \times 4 \times 7 \times 6 \times 5 \times 4 \times 3 = 90{,}720 \text{ arrangements} = 9 × 4 × 7 × 6 × 5 × 4 × 3 = 90 , 720 arrangements
