Respuesta :
Answer: 90,720
Explanation:
1) Assume initially the all the letters are different.
2) The number of ways how you can arrange 9 letters is:
P(9) = 9! = 362,880 different arrangements
3) Now take into account tha the letter n is repeated so half of the arragements are repeated.
4) The same happens with the letter e.
5) That means that you have to divide by 2 two times to get the number of differente arrays:
=> 362,880 / (2 * 2) = 90,720
Explanation:
1) Assume initially the all the letters are different.
2) The number of ways how you can arrange 9 letters is:
P(9) = 9! = 362,880 different arrangements
3) Now take into account tha the letter n is repeated so half of the arragements are repeated.
4) The same happens with the letter e.
5) That means that you have to divide by 2 two times to get the number of differente arrays:
=> 362,880 / (2 * 2) = 90,720
The number of different ways in which the letter of “PERSONNEL” can be arranged is [tex]\boxed{\bf 90,720}[/tex].
Further explanation:
A permutation is an arrangement of objects in their order in which we find number of ways that can occur in a period of time.
If all the letters are different then the number of arrangement can be written as follows:
[tex]\boxed{\begin{aligned}P(n)&=n!\\&=n\cdot (n-1)\cdot (n-2)...2\cdot 1\end{aligned}}[/tex]
The number of different permutation can be calculated as follows:
[tex]\boxed{p(n)=\dfrac{n(n-1)(n-2)...2\cdot 1}{n_{1}!\cdot n_{2}!}}[/tex]
Here, [tex]n_{1}[/tex] and [tex]n_{2}[/tex] are the different kind of object such that [tex]n_{1}+n_{2}+n_{3}+\cdot \cdot+n_{k}=n[/tex].
Given:
The given word is “PERSONNEL”.
Calculation:
Step 1:
First consider that all the letters are different i.e, repetition of letter is allowed.
The given word “PERSONNEL” has [tex]9[/tex] letters.
Therefore, the number of ways for arranging nine letters can be calculated as follows:
[tex]\begin{aligned}P(9)&=9\cdot (9-1)\cdot (9-2)\cdot \cdot 2\cdot 1\\&=9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot3 \cdot 2\cdot 1\\&=362,880\end{aligned}[/tex]
Therefore, the different arrangements of the given word is [tex]362,880[/tex].
Step 2:
Since, the letter N and E is repeated two times in given word “PERSONNEL” then the half of the arrangement is repeated of these two letters.
Use the formula for finding the number of distinguishable permutation. It can be calculated as follows:
[tex]\begin{aligned}\text{Number of permutation}&=\dfrac{362880}{2!\cdot 2!}\\&=\drac{362880}{4}\\&=90,720\end{aligned}[/tex]
The number of different ways that the letter of “PERSONNEL” can be arranged is [tex]90,720[/tex].
Thus, the number of different ways that the letter of “PERSONNEL” can be arranged is [tex]\boxed{\bf 90,720}[/tex].
Learn more:
1. Learn more about the representation of the graph https://brainly.com/question/2491745
2. Learn more about the graph of the quadratic function https://brainly.com/question/2334270
3. Learn more about graphed below of the function https://brainly.com/question/9590016
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Permutation
Keywords: Permutation, combination, personnel, counting, arrangement, different ways, order, letters, different ways, repetition, arranged objects, distinguishable permutation, counting problem.
