Math Problem Statement

6Pn=360

Solution

To solve the equation 6Pn=3606Pn = 360, we need to understand that 6Pn6Pn represents the number of permutations of 6 objects taken nn at a time.

The formula for permutations is given by:

6Pn=6!(6n)!6Pn = \frac{6!}{(6-n)!}

We are given that 6Pn=3606Pn = 360, so:

6!(6n)!=360\frac{6!}{(6-n)!} = 360

Let's calculate 6!6! (which is the factorial of 6):

6!=6×5×4×3×2×1=7206! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

Now, we can set up the equation:

720(6n)!=360\frac{720}{(6-n)!} = 360

Next, solve for (6n)!(6-n)!:

(6n)!=720360=2(6-n)! = \frac{720}{360} = 2

Now, we need to find which factorial equals 2:

1!=1,2!=21! = 1, \quad 2! = 2

This means (6n)!=2(6-n)! = 2 implies 6n=26-n = 2.

So:

6n=26 - n = 2

Now, solve for nn:

n=62=4n = 6 - 2 = 4

Final Answer:

The value of nn is 4.

Would you like to see more details or have any other questions? Here are some related questions you might find interesting:

  1. What would the equation look like if 5Pn=1205Pn = 120?
  2. How do you calculate permutations for larger values of nn?
  3. How are permutations different from combinations?
  4. Can you use permutations in probability calculations?
  5. What if the permutation was 7Pn=25207Pn = 2520? How would you solve it?

Tip: When solving permutation problems, breaking down the factorials into smaller components can make it easier to find the solution.

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Math Problem Analysis

Mathematical Concepts

Permutations
Factorials

Formulas

Permutations formula: nPr = n! / (n-r)!

Theorems

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Suitable Grade Level

High School